Properties

Label 2-624-1.1-c3-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·5-s − 20·7-s + 9·9-s − 24·11-s + 13·13-s + 18·15-s − 30·17-s + 16·19-s − 60·21-s + 72·23-s − 89·25-s + 27·27-s − 282·29-s − 164·31-s − 72·33-s − 120·35-s + 110·37-s + 39·39-s − 126·41-s − 164·43-s + 54·45-s + 204·47-s + 57·49-s − 90·51-s − 738·53-s − 144·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.536·5-s − 1.07·7-s + 1/3·9-s − 0.657·11-s + 0.277·13-s + 0.309·15-s − 0.428·17-s + 0.193·19-s − 0.623·21-s + 0.652·23-s − 0.711·25-s + 0.192·27-s − 1.80·29-s − 0.950·31-s − 0.379·33-s − 0.579·35-s + 0.488·37-s + 0.160·39-s − 0.479·41-s − 0.581·43-s + 0.178·45-s + 0.633·47-s + 0.166·49-s − 0.247·51-s − 1.91·53-s − 0.353·55-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: \(1\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 6 T + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
17 \( 1 + 30 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 + 282 T + p^{3} T^{2} \)
31 \( 1 + 164 T + p^{3} T^{2} \)
37 \( 1 - 110 T + p^{3} T^{2} \)
41 \( 1 + 126 T + p^{3} T^{2} \)
43 \( 1 + 164 T + p^{3} T^{2} \)
47 \( 1 - 204 T + p^{3} T^{2} \)
53 \( 1 + 738 T + p^{3} T^{2} \)
59 \( 1 + 120 T + p^{3} T^{2} \)
61 \( 1 - 614 T + p^{3} T^{2} \)
67 \( 1 + 848 T + p^{3} T^{2} \)
71 \( 1 + 132 T + p^{3} T^{2} \)
73 \( 1 - 218 T + p^{3} T^{2} \)
79 \( 1 - 1096 T + p^{3} T^{2} \)
83 \( 1 + 552 T + p^{3} T^{2} \)
89 \( 1 - 210 T + p^{3} T^{2} \)
97 \( 1 + 1726 T + p^{3} T^{2} \)
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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

−9.546900210821679286700026770348, −9.187688498005841625032270074148, −8.011040252800378376925797903022, −7.12174286609688391661393449228, −6.17076849160542616540780103231, −5.25619123462550059734827206700, −3.85135646283813820969142191764, −2.94088293512803917780927237310, −1.80684859441965568236244382169, 0, 1.80684859441965568236244382169, 2.94088293512803917780927237310, 3.85135646283813820969142191764, 5.25619123462550059734827206700, 6.17076849160542616540780103231, 7.12174286609688391661393449228, 8.011040252800378376925797903022, 9.187688498005841625032270074148, 9.546900210821679286700026770348

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