Properties

Label 2-4015-1.1-c1-0-62
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  + 0.0631·2-s + 0.00106·3-s − 1.99·4-s + 5-s + 6.72e−5·6-s + 3.26·7-s − 0.252·8-s − 2.99·9-s + 0.0631·10-s − 11-s − 0.00212·12-s + 0.254·13-s + 0.206·14-s + 0.00106·15-s + 3.97·16-s − 5.08·17-s − 0.189·18-s − 0.0662·19-s − 1.99·20-s + 0.00347·21-s − 0.0631·22-s + 3.74·23-s − 0.000268·24-s + 25-s + 0.0160·26-s − 0.00638·27-s − 6.51·28-s + ⋯
L(s)  = 1  + 0.0446·2-s + 0.000614·3-s − 0.998·4-s + 0.447·5-s + 2.74e−5·6-s + 1.23·7-s − 0.0892·8-s − 0.999·9-s + 0.0199·10-s − 0.301·11-s − 0.000613·12-s + 0.0707·13-s + 0.0550·14-s + 0.000274·15-s + 0.994·16-s − 1.23·17-s − 0.0446·18-s − 0.0152·19-s − 0.446·20-s + 0.000757·21-s − 0.0134·22-s + 0.781·23-s − 5.48e − 5·24-s + 0.200·25-s + 0.00315·26-s − 0.00122·27-s − 1.23·28-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

\(L(1)\) \(\approx\) \(1.561515697\)
\(L(\frac12)\) \(\approx\) \(1.561515697\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.0631T + 2T^{2} \)
3 \( 1 - 0.00106T + 3T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
13 \( 1 - 0.254T + 13T^{2} \)
17 \( 1 + 5.08T + 17T^{2} \)
19 \( 1 + 0.0662T + 19T^{2} \)
23 \( 1 - 3.74T + 23T^{2} \)
29 \( 1 - 3.06T + 29T^{2} \)
31 \( 1 - 0.0370T + 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 + 6.11T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 - 7.27T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 - 9.97T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + 9.81T + 97T^{2} \)
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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

−8.458101891831094831071300229514, −8.048616745416659883992213431177, −6.98817579967869260654458097447, −6.06997638405097095050749223871, −5.23274262149616894671306879021, −4.87241906308266238771863565011, −4.01492432293780717914399198423, −2.91051467284936564615218854107, −1.98001539563908026201202655664, −0.71845919690758723132680107319, 0.71845919690758723132680107319, 1.98001539563908026201202655664, 2.91051467284936564615218854107, 4.01492432293780717914399198423, 4.87241906308266238771863565011, 5.23274262149616894671306879021, 6.06997638405097095050749223871, 6.98817579967869260654458097447, 8.048616745416659883992213431177, 8.458101891831094831071300229514

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