Properties

Label 2-6015-1.1-c1-0-153
Degree $2$
Conductor $6015$
Sign $-1$
Analytic cond. $48.0300$
Root an. cond. $6.93036$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.462·2-s − 3-s − 1.78·4-s + 5-s + 0.462·6-s − 1.50·7-s + 1.75·8-s + 9-s − 0.462·10-s + 2.86·11-s + 1.78·12-s + 0.496·13-s + 0.697·14-s − 15-s + 2.76·16-s − 5.20·17-s − 0.462·18-s + 5.20·19-s − 1.78·20-s + 1.50·21-s − 1.32·22-s − 4.53·23-s − 1.75·24-s + 25-s − 0.229·26-s − 27-s + 2.69·28-s + ⋯
L(s)  = 1  − 0.326·2-s − 0.577·3-s − 0.893·4-s + 0.447·5-s + 0.188·6-s − 0.569·7-s + 0.618·8-s + 0.333·9-s − 0.146·10-s + 0.864·11-s + 0.515·12-s + 0.137·13-s + 0.186·14-s − 0.258·15-s + 0.690·16-s − 1.26·17-s − 0.108·18-s + 1.19·19-s − 0.399·20-s + 0.329·21-s − 0.282·22-s − 0.944·23-s − 0.357·24-s + 0.200·25-s − 0.0450·26-s − 0.192·27-s + 0.508·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $-1$
Analytic conductor: \(48.0300\)
Root analytic conductor: \(6.93036\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6015,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
401 \( 1 - T \)
good2 \( 1 + 0.462T + 2T^{2} \)
7 \( 1 + 1.50T + 7T^{2} \)
11 \( 1 - 2.86T + 11T^{2} \)
13 \( 1 - 0.496T + 13T^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 - 5.20T + 19T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 9.30T + 37T^{2} \)
41 \( 1 + 5.02T + 41T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 0.103T + 53T^{2} \)
59 \( 1 - 6.16T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 3.82T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 4.89T + 89T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−7.62319268622386477505761633585, −7.08183344546127226762552507619, −6.13862797006490754533870675207, −5.72676947002131709743375835489, −4.82345818774522614861573121259, −4.10161277673723649975990992658, −3.43754273787159629892368740185, −2.09652391966238402905415544635, −1.10119683400929135067614076832, 0, 1.10119683400929135067614076832, 2.09652391966238402905415544635, 3.43754273787159629892368740185, 4.10161277673723649975990992658, 4.82345818774522614861573121259, 5.72676947002131709743375835489, 6.13862797006490754533870675207, 7.08183344546127226762552507619, 7.62319268622386477505761633585

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