Properties

Label 2-2415-1.1-c1-0-51
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  − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 6·13-s + 14-s + 15-s − 16-s − 2·17-s − 18-s + 20-s + 21-s − 4·22-s − 23-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-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 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.689826211509155114573979136532, −7.81761317378765409612184255837, −7.11221641251139968421919176216, −6.46757385686276194163119881194, −5.34067514323999524648878050982, −4.49259103823795207312367096051, −3.96058678177192292607993886295, −2.52879661840008857202697720900, −1.11290859636845340340941393854, 0, 1.11290859636845340340941393854, 2.52879661840008857202697720900, 3.96058678177192292607993886295, 4.49259103823795207312367096051, 5.34067514323999524648878050982, 6.46757385686276194163119881194, 7.11221641251139968421919176216, 7.81761317378765409612184255837, 8.689826211509155114573979136532

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