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Properties

Label 1-2415-2415.2144-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.0331 + 0.999i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.142 + 0.989i)8-s + (0.0475 + 0.998i)11-s + (0.654 + 0.755i)13-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (−0.580 + 0.814i)19-s + 22-s + (0.786 − 0.618i)26-s + (−0.415 + 0.909i)29-s + (−0.786 − 0.618i)31-s + (0.235 − 0.971i)32-s + (−0.841 + 0.540i)34-s + (0.723 + 0.690i)37-s + (0.786 + 0.618i)38-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.142 + 0.989i)8-s + (0.0475 + 0.998i)11-s + (0.654 + 0.755i)13-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (−0.580 + 0.814i)19-s + 22-s + (0.786 − 0.618i)26-s + (−0.415 + 0.909i)29-s + (−0.786 − 0.618i)31-s + (0.235 − 0.971i)32-s + (−0.841 + 0.540i)34-s + (0.723 + 0.690i)37-s + (0.786 + 0.618i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0331 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0331 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.0331 + 0.999i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (2144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.0331 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3439438811 + 0.3555526633i\)
\(L(\frac12)\) \(\approx\) \(0.3439438811 + 0.3555526633i\)
\(L(1)\) \(\approx\) \(0.7849999599 - 0.2330190912i\)
\(L(1)\) \(\approx\) \(0.7849999599 - 0.2330190912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (0.723 + 0.690i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.327 + 0.945i)T \)
59 \( 1 + (-0.981 + 0.189i)T \)
61 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.995 + 0.0950i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.3567355270446941081423960788, −18.38867525613523250322514016817, −17.83453109901636577843446748974, −17.17093924116582012544960975674, −16.39638334022001971044893042965, −15.8066763347212064598770789482, −15.112337029117025355834264306514, −14.487435728787289026721043256978, −13.5692134226654194195304550357, −13.10623865728057434642255395810, −12.44827299465659202651412599955, −11.085759481801197654031178507732, −10.78145073014562164678213295368, −9.566934470651302764813237891706, −8.91759784214375420428982713914, −8.21207900455397373702590839070, −7.65014509310744053973577925406, −6.52137939697395387284649288670, −6.08152896368896629345533554503, −5.33595959336481029935151321255, −4.340679592669100570297288944744, −3.66169322803994114902765659509, −2.71580356444449413271646330819, −1.29284756830708254550134704535, −0.16597812947291619894508104850, 1.35643842814754526491159433852, 1.994851019444947972001405754560, 2.879876372780903993395228086410, 3.944608188326482284760543917547, 4.41440957029830525805566339535, 5.32067733924341344762022757280, 6.26282557206327176768542639680, 7.21927649882945387078797688094, 8.08788452529187922554027418638, 9.12157336950470821449215038316, 9.37745485217381685710585442819, 10.428706953858056405044735702999, 10.96211343395720247825763426262, 11.83179070275169477314104243909, 12.33318895525607861342607470421, 13.21393327562948631282890422489, 13.72730315472666614095737155338, 14.6561161728048661205566762336, 15.14866987456384183227814158449, 16.35141442341691999696202994829, 16.89878431775089360910964521199, 17.9237440362133370656604511091, 18.34621718729594046177460916402, 18.97497480177286702186360347217, 19.91191853298770813161717201439

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