Properties

Label 6-18e6-1.1-c7e3-0-0
Degree $6$
Conductor $34012224$
Sign $-1$
Analytic cond. $1.03682\times 10^{6}$
Root an. cond. $10.0604$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 57·5-s + 114·7-s + 438·11-s + 15·13-s − 1.23e3·17-s − 8.52e3·19-s + 8.20e3·23-s − 1.18e5·25-s + 2.28e4·29-s + 1.61e4·31-s − 6.49e3·35-s − 1.04e5·37-s + 2.13e5·41-s − 8.48e4·43-s − 3.73e5·47-s − 1.31e6·49-s − 1.03e5·53-s − 2.49e4·55-s + 1.20e6·59-s − 7.90e5·61-s − 855·65-s − 1.99e6·67-s − 1.12e6·71-s − 2.01e6·73-s + 4.99e4·77-s − 5.11e6·79-s − 8.40e6·83-s + ⋯
L(s)  = 1  − 0.203·5-s + 0.125·7-s + 0.0992·11-s + 0.00189·13-s − 0.0611·17-s − 0.285·19-s + 0.140·23-s − 1.51·25-s + 0.173·29-s + 0.0975·31-s − 0.0256·35-s − 0.339·37-s + 0.484·41-s − 0.162·43-s − 0.524·47-s − 1.60·49-s − 0.0957·53-s − 0.0202·55-s + 0.761·59-s − 0.445·61-s − 0.000386·65-s − 0.808·67-s − 0.371·71-s − 0.605·73-s + 0.0124·77-s − 1.16·79-s − 1.61·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(34012224\)    =    \(2^{6} \cdot 3^{12}\)
Sign: $-1$
Analytic conductor: \(1.03682\times 10^{6}\)
Root analytic conductor: \(10.0604\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 34012224,\ (\ :7/2, 7/2, 7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$S_4\times C_2$ \( 1 + 57 T + 24366 p T^{2} + 394617 p^{2} T^{3} + 24366 p^{8} T^{4} + 57 p^{14} T^{5} + p^{21} T^{6} \)
7$S_4\times C_2$ \( 1 - 114 T + 1331133 T^{2} - 612806780 T^{3} + 1331133 p^{7} T^{4} - 114 p^{14} T^{5} + p^{21} T^{6} \)
11$S_4\times C_2$ \( 1 - 438 T + 47937201 T^{2} - 25437983172 T^{3} + 47937201 p^{7} T^{4} - 438 p^{14} T^{5} + p^{21} T^{6} \)
13$S_4\times C_2$ \( 1 - 15 T + 159776286 T^{2} - 9954578135 T^{3} + 159776286 p^{7} T^{4} - 15 p^{14} T^{5} + p^{21} T^{6} \)
17$S_4\times C_2$ \( 1 + 1239 T + 465599598 T^{2} + 5498376714195 T^{3} + 465599598 p^{7} T^{4} + 1239 p^{14} T^{5} + p^{21} T^{6} \)
19$S_4\times C_2$ \( 1 + 8526 T + 59517411 p T^{2} - 11055844705484 T^{3} + 59517411 p^{8} T^{4} + 8526 p^{14} T^{5} + p^{21} T^{6} \)
23$S_4\times C_2$ \( 1 - 8202 T + 7915582797 T^{2} - 92377955346060 T^{3} + 7915582797 p^{7} T^{4} - 8202 p^{14} T^{5} + p^{21} T^{6} \)
29$S_4\times C_2$ \( 1 - 22839 T + 16307508894 T^{2} + 708273351805521 T^{3} + 16307508894 p^{7} T^{4} - 22839 p^{14} T^{5} + p^{21} T^{6} \)
31$S_4\times C_2$ \( 1 - 16188 T + 19853899341 T^{2} - 6403969871615432 T^{3} + 19853899341 p^{7} T^{4} - 16188 p^{14} T^{5} + p^{21} T^{6} \)
37$S_4\times C_2$ \( 1 + 104745 T + 101471603334 T^{2} + 35277380195298145 T^{3} + 101471603334 p^{7} T^{4} + 104745 p^{14} T^{5} + p^{21} T^{6} \)
41$S_4\times C_2$ \( 1 - 213738 T + 503336992551 T^{2} - 65324719417880652 T^{3} + 503336992551 p^{7} T^{4} - 213738 p^{14} T^{5} + p^{21} T^{6} \)
43$S_4\times C_2$ \( 1 + 84810 T + 784322721681 T^{2} + 43003546629647740 T^{3} + 784322721681 p^{7} T^{4} + 84810 p^{14} T^{5} + p^{21} T^{6} \)
47$S_4\times C_2$ \( 1 + 373680 T + 708326752029 T^{2} + 572673134531602080 T^{3} + 708326752029 p^{7} T^{4} + 373680 p^{14} T^{5} + p^{21} T^{6} \)
53$S_4\times C_2$ \( 1 + 103782 T + 3345290776707 T^{2} + 208121271067524900 T^{3} + 3345290776707 p^{7} T^{4} + 103782 p^{14} T^{5} + p^{21} T^{6} \)
59$S_4\times C_2$ \( 1 - 1201404 T + 6732186959769 T^{2} - 6072801756720236904 T^{3} + 6732186959769 p^{7} T^{4} - 1201404 p^{14} T^{5} + p^{21} T^{6} \)
61$S_4\times C_2$ \( 1 + 790545 T + 8461371031278 T^{2} + 4258902460629749065 T^{3} + 8461371031278 p^{7} T^{4} + 790545 p^{14} T^{5} + p^{21} T^{6} \)
67$S_4\times C_2$ \( 1 + 1991406 T + 15262062748233 T^{2} + 19818531193008129460 T^{3} + 15262062748233 p^{7} T^{4} + 1991406 p^{14} T^{5} + p^{21} T^{6} \)
71$S_4\times C_2$ \( 1 + 1121478 T + 19822640020701 T^{2} + 24797759778346173972 T^{3} + 19822640020701 p^{7} T^{4} + 1121478 p^{14} T^{5} + p^{21} T^{6} \)
73$S_4\times C_2$ \( 1 + 2010963 T + 19031868700062 T^{2} + 27441661678448920855 T^{3} + 19031868700062 p^{7} T^{4} + 2010963 p^{14} T^{5} + p^{21} T^{6} \)
79$S_4\times C_2$ \( 1 + 5111130 T + 29824596730437 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + 29824596730437 p^{7} T^{4} + 5111130 p^{14} T^{5} + p^{21} T^{6} \)
83$S_4\times C_2$ \( 1 + 8408880 T + 76111620401121 T^{2} + \)\(39\!\cdots\!20\)\( T^{3} + 76111620401121 p^{7} T^{4} + 8408880 p^{14} T^{5} + p^{21} T^{6} \)
89$S_4\times C_2$ \( 1 + 3412959 T + 31069479367014 T^{2} + \)\(47\!\cdots\!99\)\( T^{3} + 31069479367014 p^{7} T^{4} + 3412959 p^{14} T^{5} + p^{21} T^{6} \)
97$S_4\times C_2$ \( 1 + 112710 p T + 213340799112639 T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + 213340799112639 p^{7} T^{4} + 112710 p^{15} T^{5} + p^{21} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.687034724832136295021479973052, −9.425586019723841194251050735629, −8.850621014075768810464610618176, −8.814634663356221357643755924981, −8.164596815719529623819926979298, −8.065440380681316760801205109475, −7.985154717112168832139738619044, −7.14168667747306386435222280838, −7.06928341056278331503039881489, −7.00539404248681395581140491347, −6.13463544453720812771588542360, −6.06679231533844044187836819074, −5.85885625989694821215590399018, −5.28284129137038595256133929337, −4.80806391014058218658991616530, −4.73902386021434899162744823465, −4.13684931371878675010676464469, −3.72913859498138441384514539774, −3.68052493033001903815020547221, −2.93499201275500092111045302742, −2.58566011455571719824600531351, −2.36956683465844057940237776856, −1.51733974237784045653681015420, −1.44976546787586841530459266000, −1.10801688723869040725895832245, 0, 0, 0, 1.10801688723869040725895832245, 1.44976546787586841530459266000, 1.51733974237784045653681015420, 2.36956683465844057940237776856, 2.58566011455571719824600531351, 2.93499201275500092111045302742, 3.68052493033001903815020547221, 3.72913859498138441384514539774, 4.13684931371878675010676464469, 4.73902386021434899162744823465, 4.80806391014058218658991616530, 5.28284129137038595256133929337, 5.85885625989694821215590399018, 6.06679231533844044187836819074, 6.13463544453720812771588542360, 7.00539404248681395581140491347, 7.06928341056278331503039881489, 7.14168667747306386435222280838, 7.985154717112168832139738619044, 8.065440380681316760801205109475, 8.164596815719529623819926979298, 8.814634663356221357643755924981, 8.850621014075768810464610618176, 9.425586019723841194251050735629, 9.687034724832136295021479973052

Graph of the $Z$-function along the critical line