Properties

Label 6-4600e3-1.1-c1e3-0-4
Degree $6$
Conductor $97336000000$
Sign $-1$
Analytic cond. $49556.9$
Root an. cond. $6.06062$
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 − 2·7-s + 9-s + 7·11-s + 13-s − 10·17-s − 13·19-s + 2·21-s − 3·23-s + 27-s + 13·29-s − 8·31-s − 7·33-s − 5·37-s − 39-s + 8·41-s − 24·43-s − 2·47-s − 9·49-s + 10·51-s − 53-s + 13·57-s − 17·59-s + 13·61-s − 2·63-s − 5·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 2.11·11-s + 0.277·13-s − 2.42·17-s − 2.98·19-s + 0.436·21-s − 0.625·23-s + 0.192·27-s + 2.41·29-s − 1.43·31-s − 1.21·33-s − 0.821·37-s − 0.160·39-s + 1.24·41-s − 3.65·43-s − 0.291·47-s − 9/7·49-s + 1.40·51-s − 0.137·53-s + 1.72·57-s − 2.21·59-s + 1.66·61-s − 0.251·63-s − 0.610·67-s + 0.361·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{6} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(49556.9\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 5^{6} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
23$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 + T - 2 T^{3} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 2 T + 13 T^{2} + 27 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 7 T + 40 T^{2} - 140 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - T + 16 T^{2} + 24 T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 10 T + 61 T^{2} + 257 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 13 T + 90 T^{2} + 432 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 13 T + 113 T^{2} - 678 T^{3} + 113 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 8 T + 105 T^{2} + 495 T^{3} + 105 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 5 T + 19 T^{2} - 126 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{3} \)
47$S_4\times C_2$ \( 1 + 2 T + 49 T^{2} - 212 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + T + 59 T^{2} + 406 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 17 T + 243 T^{2} + 2042 T^{3} + 243 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 13 T + 132 T^{2} - 866 T^{3} + 132 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 5 T + 109 T^{2} + 174 T^{3} + 109 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 22 T + 309 T^{2} + 2899 T^{3} + 309 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 12 T + 43 T^{2} - 368 T^{3} + 43 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 4 T + 93 T^{2} + 120 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 15 T + 289 T^{2} - 2454 T^{3} + 289 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 3 T + 210 T^{2} - 632 T^{3} + 210 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.87514941104700597229788115631, −7.19604543688968685088377875065, −7.16064094146726709545067251720, −6.74815194530357467066746245574, −6.71658919030604955381364347196, −6.45486661363908126629452780591, −6.43271059024484446726529033762, −6.03049509525168221404093351821, −6.02654374442356144941112713072, −5.77425889653177670937104707541, −5.02370206111727381503902487309, −4.85190039839795922709094010303, −4.68813612639561842765545932106, −4.45661255648631534006281559936, −4.35890812596204029786926128855, −3.84352366124386697701380113780, −3.77347063404550293108797487320, −3.35330869837054529412550564426, −3.24817954856466083233044248048, −2.51965338054142476582722345748, −2.51563998092707630371367353270, −1.96450535390817251664402918682, −1.80847215035675707208142687497, −1.27600359040301375644315114772, −1.24989929006281971613781897180, 0, 0, 0, 1.24989929006281971613781897180, 1.27600359040301375644315114772, 1.80847215035675707208142687497, 1.96450535390817251664402918682, 2.51563998092707630371367353270, 2.51965338054142476582722345748, 3.24817954856466083233044248048, 3.35330869837054529412550564426, 3.77347063404550293108797487320, 3.84352366124386697701380113780, 4.35890812596204029786926128855, 4.45661255648631534006281559936, 4.68813612639561842765545932106, 4.85190039839795922709094010303, 5.02370206111727381503902487309, 5.77425889653177670937104707541, 6.02654374442356144941112713072, 6.03049509525168221404093351821, 6.43271059024484446726529033762, 6.45486661363908126629452780591, 6.71658919030604955381364347196, 6.74815194530357467066746245574, 7.16064094146726709545067251720, 7.19604543688968685088377875065, 7.87514941104700597229788115631

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