Properties

Label 2-2415-1.1-c3-0-197
Degree $2$
Conductor $2415$
Sign $-1$
Analytic cond. $142.489$
Root an. cond. $11.9369$
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 − 8·4-s − 5·5-s + 7·7-s + 9·9-s + 12·11-s − 24·12-s − 42·13-s − 15·15-s + 64·16-s + 26·17-s − 26·19-s + 40·20-s + 21·21-s + 23·23-s + 25·25-s + 27·27-s − 56·28-s − 82·29-s − 64·31-s + 36·33-s − 35·35-s − 72·36-s − 256·37-s − 126·39-s + 126·41-s + 508·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.328·11-s − 0.577·12-s − 0.896·13-s − 0.258·15-s + 16-s + 0.370·17-s − 0.313·19-s + 0.447·20-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.525·29-s − 0.370·31-s + 0.189·33-s − 0.169·35-s − 1/3·36-s − 1.13·37-s − 0.517·39-s + 0.479·41-s + 1.80·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(142.489\)
Root analytic conductor: \(11.9369\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2415,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 - p T \)
5 \( 1 + p T \)
7 \( 1 - p T \)
23 \( 1 - p T \)
good2 \( 1 + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 42 T + p^{3} T^{2} \)
17 \( 1 - 26 T + p^{3} T^{2} \)
19 \( 1 + 26 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 + 64 T + p^{3} T^{2} \)
37 \( 1 + 256 T + p^{3} T^{2} \)
41 \( 1 - 126 T + p^{3} T^{2} \)
43 \( 1 - 508 T + p^{3} T^{2} \)
47 \( 1 - 4 T + p^{3} T^{2} \)
53 \( 1 + 102 T + p^{3} T^{2} \)
59 \( 1 + 478 T + p^{3} T^{2} \)
61 \( 1 - 462 T + p^{3} T^{2} \)
67 \( 1 - 724 T + p^{3} T^{2} \)
71 \( 1 + 90 T + p^{3} T^{2} \)
73 \( 1 + 82 T + p^{3} T^{2} \)
79 \( 1 - 798 T + p^{3} T^{2} \)
83 \( 1 + 748 T + p^{3} T^{2} \)
89 \( 1 - 1122 T + p^{3} T^{2} \)
97 \( 1 + 1452 T + p^{3} T^{2} \)
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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

−8.200289114431233876297739417393, −7.67616048263653292634182422222, −6.90834781986398171924302353013, −5.67344209073177794106420747519, −4.90984358455714755346091494091, −4.14713625880723693740180703985, −3.47508077383348727055863122344, −2.35663240155113101346399001756, −1.13998601751280699736553606232, 0, 1.13998601751280699736553606232, 2.35663240155113101346399001756, 3.47508077383348727055863122344, 4.14713625880723693740180703985, 4.90984358455714755346091494091, 5.67344209073177794106420747519, 6.90834781986398171924302353013, 7.67616048263653292634182422222, 8.200289114431233876297739417393

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