Properties

Label 2-2415-1.1-c1-0-76
Degree $2$
Conductor $2415$
Sign $-1$
Analytic cond. $19.2838$
Root an. cond. $4.39134$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s + 7-s + 9-s + 3·11-s − 2·12-s − 4·13-s − 15-s + 4·16-s − 6·17-s − 19-s + 2·20-s + 21-s − 23-s + 25-s + 27-s − 2·28-s + 6·29-s + 8·31-s + 3·33-s − 35-s − 2·36-s − 10·37-s − 4·39-s − 3·41-s − 10·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s − 0.258·15-s + 16-s − 1.45·17-s − 0.229·19-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 + 1.11·29-s + 1.43·31-s + 0.522·33-s − 0.169·35-s − 1/3·36-s − 1.64·37-s − 0.640·39-s − 0.468·41-s − 1.52·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(19.2838\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2415} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2415,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.485749227808954796631523822352, −8.160625928367124194564103725231, −7.04880326351232874256217328676, −6.44820708004920735214806528518, −5.00010326554341053545442709036, −4.58491344295942601284268345621, −3.81274100836837068759670803629, −2.78979023966689197511354892893, −1.54049047545509520664186580686, 0, 1.54049047545509520664186580686, 2.78979023966689197511354892893, 3.81274100836837068759670803629, 4.58491344295942601284268345621, 5.00010326554341053545442709036, 6.44820708004920735214806528518, 7.04880326351232874256217328676, 8.160625928367124194564103725231, 8.485749227808954796631523822352

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