Properties

Label 2-4015-1.1-c1-0-133
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0204·2-s + 2.32·3-s − 1.99·4-s + 5-s + 0.0475·6-s + 2.78·7-s − 0.0818·8-s + 2.39·9-s + 0.0204·10-s − 11-s − 4.64·12-s + 4.04·13-s + 0.0570·14-s + 2.32·15-s + 3.99·16-s + 4.96·17-s + 0.0490·18-s + 5.19·19-s − 1.99·20-s + 6.46·21-s − 0.0204·22-s − 4.50·23-s − 0.190·24-s + 25-s + 0.0828·26-s − 1.40·27-s − 5.57·28-s + ⋯
L(s)  = 1  + 0.0144·2-s + 1.34·3-s − 0.999·4-s + 0.447·5-s + 0.0194·6-s + 1.05·7-s − 0.0289·8-s + 0.798·9-s + 0.00647·10-s − 0.301·11-s − 1.34·12-s + 1.12·13-s + 0.0152·14-s + 0.599·15-s + 0.999·16-s + 1.20·17-s + 0.0115·18-s + 1.19·19-s − 0.447·20-s + 1.41·21-s − 0.00436·22-s − 0.938·23-s − 0.0388·24-s + 0.200·25-s + 0.0162·26-s − 0.270·27-s − 1.05·28-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: \(0\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.390651518\)
\(L(\frac12)\) \(\approx\) \(3.390651518\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.0204T + 2T^{2} \)
3 \( 1 - 2.32T + 3T^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
13 \( 1 - 4.04T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 + 4.50T + 23T^{2} \)
29 \( 1 + 4.85T + 29T^{2} \)
31 \( 1 - 4.18T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 - 3.05T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 8.87T + 67T^{2} \)
71 \( 1 - 3.10T + 71T^{2} \)
79 \( 1 + 1.96T + 79T^{2} \)
83 \( 1 - 7.53T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 0.804T + 97T^{2} \)
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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

−8.326623440769239436600487267156, −7.960506571521640728602987888716, −7.47191489505469161520020424860, −5.97900724083525805679715657904, −5.44078596455511567938476153806, −4.57528963912409998555537670746, −3.67791865498517867242921777092, −3.14850294813541706609175742365, −1.93391643178571195008551417108, −1.10010122891048865475388457280, 1.10010122891048865475388457280, 1.93391643178571195008551417108, 3.14850294813541706609175742365, 3.67791865498517867242921777092, 4.57528963912409998555537670746, 5.44078596455511567938476153806, 5.97900724083525805679715657904, 7.47191489505469161520020424860, 7.960506571521640728602987888716, 8.326623440769239436600487267156

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