Properties

Label 6-2960e3-1.1-c1e3-0-3
Degree $6$
Conductor $25934336000$
Sign $-1$
Analytic cond. $13204.0$
Root an. cond. $4.86165$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 7-s − 2·9-s − 5·11-s − 3·15-s − 8·19-s − 21-s − 12·23-s + 6·25-s − 3·27-s − 4·29-s − 6·31-s + 5·33-s + 3·35-s + 3·37-s − 41-s − 10·43-s − 6·45-s − 3·47-s − 14·49-s + 11·53-s − 15·55-s + 8·57-s + 4·59-s − 10·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.774·15-s − 1.83·19-s − 0.218·21-s − 2.50·23-s + 6/5·25-s − 0.577·27-s − 0.742·29-s − 1.07·31-s + 0.870·33-s + 0.507·35-s + 0.493·37-s − 0.156·41-s − 1.52·43-s − 0.894·45-s − 0.437·47-s − 2·49-s + 1.51·53-s − 2.02·55-s + 1.05·57-s + 0.520·59-s − 1.28·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 37^{3}\)
Sign: $-1$
Analytic conductor: \(13204.0\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 37^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
37$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 15 T^{2} - 16 T^{3} + 15 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 5 T + 3 p T^{2} + 102 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + p T^{2} )^{3} \)
17$C_2$ \( ( 1 + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 8 T + 59 T^{2} + 240 T^{3} + 59 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 + 4 T + 59 T^{2} + 200 T^{3} + 59 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 6 T + 95 T^{2} + 368 T^{3} + 95 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + T + 99 T^{2} + 102 T^{3} + 99 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 3 p T^{2} + 796 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 3 T + 23 T^{2} + 416 T^{3} + 23 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 11 T + 127 T^{2} - 714 T^{3} + 127 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 4 T + 115 T^{2} - 240 T^{3} + 115 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 10 T + 115 T^{2} + 1180 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 28 T + 395 T^{2} + 3792 T^{3} + 395 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 15 T + 245 T^{2} + 2098 T^{3} + 245 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 5 T + 99 T^{2} + 34 T^{3} + 99 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 143 T^{2} + 528 T^{3} + 143 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 5 T + 19 T^{2} + 936 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 6 T + 119 T^{2} + 1268 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{3} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.226985393869600505416713418882, −7.83799449061939219747535230609, −7.61704325784401605453653018508, −7.52819679329816425181517887959, −7.14180844127611195199551233174, −6.61555926967380592267472205265, −6.57797412407734180169563053998, −6.21051063510896026392878197772, −6.05436070894271714123132871619, −5.84757474444869880476057354559, −5.53109873355808720688838349320, −5.35341608069528347382955764945, −5.24308442386988117172680867809, −4.77704454272368536988252881158, −4.43683025583940637510222960265, −4.28305067836020154265702049463, −3.96510771611150243270936538530, −3.47414873102356570411570152867, −3.29075223151075227607716128858, −2.70432794332268591438512901533, −2.58805865444571387122063407772, −2.21582638801360755608465398356, −1.83530597956227026303118743508, −1.65187748319839035007836031412, −1.35540703183763351804611562360, 0, 0, 0, 1.35540703183763351804611562360, 1.65187748319839035007836031412, 1.83530597956227026303118743508, 2.21582638801360755608465398356, 2.58805865444571387122063407772, 2.70432794332268591438512901533, 3.29075223151075227607716128858, 3.47414873102356570411570152867, 3.96510771611150243270936538530, 4.28305067836020154265702049463, 4.43683025583940637510222960265, 4.77704454272368536988252881158, 5.24308442386988117172680867809, 5.35341608069528347382955764945, 5.53109873355808720688838349320, 5.84757474444869880476057354559, 6.05436070894271714123132871619, 6.21051063510896026392878197772, 6.57797412407734180169563053998, 6.61555926967380592267472205265, 7.14180844127611195199551233174, 7.52819679329816425181517887959, 7.61704325784401605453653018508, 7.83799449061939219747535230609, 8.226985393869600505416713418882

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