Properties

Label 2-4012-1.1-c1-0-38
Degree $2$
Conductor $4012$
Sign $-1$
Analytic cond. $32.0359$
Root an. cond. $5.66003$
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  − 2.34·3-s + 3.93·5-s − 4.81·7-s + 2.51·9-s − 2.50·11-s − 4.25·13-s − 9.22·15-s + 17-s + 7.47·19-s + 11.3·21-s − 2.47·23-s + 10.4·25-s + 1.13·27-s + 6.29·29-s + 5.98·31-s + 5.89·33-s − 18.9·35-s + 3.40·37-s + 9.98·39-s + 6.76·41-s − 3.05·43-s + 9.88·45-s − 13.5·47-s + 16.1·49-s − 2.34·51-s − 11.4·53-s − 9.86·55-s + ⋯
L(s)  = 1  − 1.35·3-s + 1.75·5-s − 1.81·7-s + 0.838·9-s − 0.756·11-s − 1.17·13-s − 2.38·15-s + 0.242·17-s + 1.71·19-s + 2.46·21-s − 0.516·23-s + 2.08·25-s + 0.219·27-s + 1.16·29-s + 1.07·31-s + 1.02·33-s − 3.19·35-s + 0.559·37-s + 1.59·39-s + 1.05·41-s − 0.465·43-s + 1.47·45-s − 1.96·47-s + 2.30·49-s − 0.328·51-s − 1.57·53-s − 1.32·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(32.0359\)
Root analytic conductor: \(5.66003\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4012,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
17 \( 1 - T \)
59 \( 1 + T \)
good3 \( 1 + 2.34T + 3T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
7 \( 1 + 4.81T + 7T^{2} \)
11 \( 1 + 2.50T + 11T^{2} \)
13 \( 1 + 4.25T + 13T^{2} \)
19 \( 1 - 7.47T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 - 3.40T + 37T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 + 3.05T + 43T^{2} \)
47 \( 1 + 13.5T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + 7.93T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 + 8.11T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 9.46T + 97T^{2} \)
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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

−7.922071598110300615738091835078, −6.95967740396312226258480851349, −6.34065365400095837644931695673, −5.98642762194462548602703367050, −5.25129341617138888318418778476, −4.73975488458831674373445516272, −3.04297878424062566804503340678, −2.65608344146016730979711990289, −1.18695606210322690576879922610, 0, 1.18695606210322690576879922610, 2.65608344146016730979711990289, 3.04297878424062566804503340678, 4.73975488458831674373445516272, 5.25129341617138888318418778476, 5.98642762194462548602703367050, 6.34065365400095837644931695673, 6.95967740396312226258480851349, 7.922071598110300615738091835078

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