Properties

Label 2-4022-1.1-c1-0-118
Degree $2$
Conductor $4022$
Sign $1$
Analytic cond. $32.1158$
Root an. cond. $5.66708$
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  + 2-s + 0.578·3-s + 4-s + 1.16·5-s + 0.578·6-s + 4.77·7-s + 8-s − 2.66·9-s + 1.16·10-s + 6.18·11-s + 0.578·12-s − 2.54·13-s + 4.77·14-s + 0.673·15-s + 16-s + 7.73·17-s − 2.66·18-s + 5.46·19-s + 1.16·20-s + 2.76·21-s + 6.18·22-s + 5.22·23-s + 0.578·24-s − 3.64·25-s − 2.54·26-s − 3.27·27-s + 4.77·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.334·3-s + 0.5·4-s + 0.520·5-s + 0.236·6-s + 1.80·7-s + 0.353·8-s − 0.888·9-s + 0.367·10-s + 1.86·11-s + 0.167·12-s − 0.704·13-s + 1.27·14-s + 0.173·15-s + 0.250·16-s + 1.87·17-s − 0.628·18-s + 1.25·19-s + 0.260·20-s + 0.602·21-s + 1.31·22-s + 1.08·23-s + 0.118·24-s − 0.729·25-s − 0.498·26-s − 0.630·27-s + 0.901·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4022\)    =    \(2 \cdot 2011\)
Sign: $1$
Analytic conductor: \(32.1158\)
Root analytic conductor: \(5.66708\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.121001805\)
\(L(\frac12)\) \(\approx\) \(5.121001805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2011 \( 1 + T \)
good3 \( 1 - 0.578T + 3T^{2} \)
5 \( 1 - 1.16T + 5T^{2} \)
7 \( 1 - 4.77T + 7T^{2} \)
11 \( 1 - 6.18T + 11T^{2} \)
13 \( 1 + 2.54T + 13T^{2} \)
17 \( 1 - 7.73T + 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 - 5.22T + 23T^{2} \)
29 \( 1 + 2.11T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 8.43T + 41T^{2} \)
43 \( 1 + 8.20T + 43T^{2} \)
47 \( 1 + 2.45T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 1.72T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 9.67T + 67T^{2} \)
71 \( 1 + 8.29T + 71T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 + 5.84T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 - 3.81T + 97T^{2} \)
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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

−8.487264081509683733732093281935, −7.46007191099096881523972631915, −7.22006590740383812062266698377, −5.91779649095284537351305747851, −5.35748786088303187347959169856, −4.90705113277298312684430421058, −3.69465963752159717077190720094, −3.19569797943358885387213949175, −1.78951796988098178256350200896, −1.42602234980072293867433460025, 1.42602234980072293867433460025, 1.78951796988098178256350200896, 3.19569797943358885387213949175, 3.69465963752159717077190720094, 4.90705113277298312684430421058, 5.35748786088303187347959169856, 5.91779649095284537351305747851, 7.22006590740383812062266698377, 7.46007191099096881523972631915, 8.487264081509683733732093281935

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