Properties

Label 2-4034-1.1-c1-0-97
Degree $2$
Conductor $4034$
Sign $-1$
Analytic cond. $32.2116$
Root an. cond. $5.67553$
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-s − 0.715·3-s + 4-s + 2.28·5-s + 0.715·6-s − 4.66·7-s − 8-s − 2.48·9-s − 2.28·10-s + 3.56·11-s − 0.715·12-s + 2.24·13-s + 4.66·14-s − 1.63·15-s + 16-s + 0.341·17-s + 2.48·18-s − 2.98·19-s + 2.28·20-s + 3.34·21-s − 3.56·22-s + 0.277·23-s + 0.715·24-s + 0.205·25-s − 2.24·26-s + 3.92·27-s − 4.66·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.413·3-s + 0.5·4-s + 1.02·5-s + 0.292·6-s − 1.76·7-s − 0.353·8-s − 0.829·9-s − 0.721·10-s + 1.07·11-s − 0.206·12-s + 0.621·13-s + 1.24·14-s − 0.421·15-s + 0.250·16-s + 0.0828·17-s + 0.586·18-s − 0.685·19-s + 0.510·20-s + 0.729·21-s − 0.761·22-s + 0.0578·23-s + 0.146·24-s + 0.0410·25-s − 0.439·26-s + 0.755·27-s − 0.882·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4034\)    =    \(2 \cdot 2017\)
Sign: $-1$
Analytic conductor: \(32.2116\)
Root analytic conductor: \(5.67553\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
2017 \( 1 + T \)
good3 \( 1 + 0.715T + 3T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 - 0.341T + 17T^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 - 0.277T + 23T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + 3.72T + 31T^{2} \)
37 \( 1 - 4.21T + 37T^{2} \)
41 \( 1 - 0.730T + 41T^{2} \)
43 \( 1 - 4.51T + 43T^{2} \)
47 \( 1 + 6.11T + 47T^{2} \)
53 \( 1 + 4.30T + 53T^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 1.09T + 73T^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 + 6.25T + 83T^{2} \)
89 \( 1 + 5.40T + 89T^{2} \)
97 \( 1 + 4.68T + 97T^{2} \)
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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

−8.373748723531008248032908744769, −7.09589154548590329075265698660, −6.44947181386862481349374664455, −6.10598172230138256643558047881, −5.53821753250745708330668725638, −4.10557733598863442688026321262, −3.20448819347477277827673218558, −2.42810645687449579687241836529, −1.22683196190181123639492662763, 0, 1.22683196190181123639492662763, 2.42810645687449579687241836529, 3.20448819347477277827673218558, 4.10557733598863442688026321262, 5.53821753250745708330668725638, 6.10598172230138256643558047881, 6.44947181386862481349374664455, 7.09589154548590329075265698660, 8.373748723531008248032908744769

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