Properties

Label 2-4015-1.1-c1-0-84
Degree $2$
Conductor $4015$
Sign $-1$
Analytic cond. $32.0599$
Root an. cond. $5.66214$
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  − 0.595·2-s − 0.809·3-s − 1.64·4-s − 5-s + 0.482·6-s − 4.26·7-s + 2.17·8-s − 2.34·9-s + 0.595·10-s + 11-s + 1.33·12-s − 3.12·13-s + 2.54·14-s + 0.809·15-s + 1.99·16-s + 2.46·17-s + 1.39·18-s + 2.20·19-s + 1.64·20-s + 3.45·21-s − 0.595·22-s − 0.564·23-s − 1.75·24-s + 25-s + 1.86·26-s + 4.32·27-s + 7.02·28-s + ⋯
L(s)  = 1  − 0.421·2-s − 0.467·3-s − 0.822·4-s − 0.447·5-s + 0.196·6-s − 1.61·7-s + 0.767·8-s − 0.781·9-s + 0.188·10-s + 0.301·11-s + 0.384·12-s − 0.866·13-s + 0.679·14-s + 0.208·15-s + 0.498·16-s + 0.596·17-s + 0.329·18-s + 0.505·19-s + 0.367·20-s + 0.753·21-s − 0.127·22-s − 0.117·23-s − 0.358·24-s + 0.200·25-s + 0.364·26-s + 0.832·27-s + 1.32·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(32.0599\)
Root analytic conductor: \(5.66214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 + 0.595T + 2T^{2} \)
3 \( 1 + 0.809T + 3T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
23 \( 1 + 0.564T + 23T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + 2.41T + 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 5.90T + 59T^{2} \)
61 \( 1 - 2.22T + 61T^{2} \)
67 \( 1 - 0.968T + 67T^{2} \)
71 \( 1 - 2.24T + 71T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 + 1.82T + 89T^{2} \)
97 \( 1 + 1.63T + 97T^{2} \)
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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

−8.188213262427111189508025069294, −7.34628802389973079148365202604, −6.69421119214160165243918181805, −5.77423786535092532136831923575, −5.22486755003551283443537793877, −4.19486199183030617410828428369, −3.45244760439671236747592324592, −2.63924439315818796752735833564, −0.858006492041433291130812407176, 0, 0.858006492041433291130812407176, 2.63924439315818796752735833564, 3.45244760439671236747592324592, 4.19486199183030617410828428369, 5.22486755003551283443537793877, 5.77423786535092532136831923575, 6.69421119214160165243918181805, 7.34628802389973079148365202604, 8.188213262427111189508025069294

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