Properties

Label 2-624-1.1-c3-0-8
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 20·5-s + 32·7-s + 9·9-s − 50·11-s − 13·13-s − 60·15-s − 30·17-s + 120·19-s + 96·21-s + 20·23-s + 275·25-s + 27·27-s + 82·29-s + 44·31-s − 150·33-s − 640·35-s − 306·37-s − 39·39-s + 108·41-s + 356·43-s − 180·45-s + 178·47-s + 681·49-s − 90·51-s + 198·53-s + 1.00e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1.72·7-s + 1/3·9-s − 1.37·11-s − 0.277·13-s − 1.03·15-s − 0.428·17-s + 1.44·19-s + 0.997·21-s + 0.181·23-s + 11/5·25-s + 0.192·27-s + 0.525·29-s + 0.254·31-s − 0.791·33-s − 3.09·35-s − 1.35·37-s − 0.160·39-s + 0.411·41-s + 1.26·43-s − 0.596·45-s + 0.552·47-s + 1.98·49-s − 0.247·51-s + 0.513·53-s + 2.45·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.929615039\)
\(L(\frac12)\) \(\approx\) \(1.929615039\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p T \)
13 \( 1 + p T \)
good5 \( 1 + 4 p T + p^{3} T^{2} \)
7 \( 1 - 32 T + p^{3} T^{2} \)
11 \( 1 + 50 T + p^{3} T^{2} \)
17 \( 1 + 30 T + p^{3} T^{2} \)
19 \( 1 - 120 T + p^{3} T^{2} \)
23 \( 1 - 20 T + p^{3} T^{2} \)
29 \( 1 - 82 T + p^{3} T^{2} \)
31 \( 1 - 44 T + p^{3} T^{2} \)
37 \( 1 + 306 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 - 356 T + p^{3} T^{2} \)
47 \( 1 - 178 T + p^{3} T^{2} \)
53 \( 1 - 198 T + p^{3} T^{2} \)
59 \( 1 + 94 T + p^{3} T^{2} \)
61 \( 1 + 62 T + p^{3} T^{2} \)
67 \( 1 - 140 T + p^{3} T^{2} \)
71 \( 1 - 778 T + p^{3} T^{2} \)
73 \( 1 - 62 T + p^{3} T^{2} \)
79 \( 1 - 1096 T + p^{3} T^{2} \)
83 \( 1 - 462 T + p^{3} T^{2} \)
89 \( 1 - 1224 T + p^{3} T^{2} \)
97 \( 1 - 614 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49393955580545483945433552139, −9.026433164144894334072160413129, −8.165543642952483528804035907523, −7.75285449659849961084871352610, −7.19643311547899476404815537721, −5.20063177596753332789385991301, −4.64627358872716445355436905449, −3.57535670729108595844303922141, −2.41556057672155807198707781396, −0.808677032725338024500324579133, 0.808677032725338024500324579133, 2.41556057672155807198707781396, 3.57535670729108595844303922141, 4.64627358872716445355436905449, 5.20063177596753332789385991301, 7.19643311547899476404815537721, 7.75285449659849961084871352610, 8.165543642952483528804035907523, 9.026433164144894334072160413129, 10.49393955580545483945433552139

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