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Properties

Label 2-1012-253.142-c0-0-1
Degree $2$
Conductor $1012$
Sign $0.211 + 0.977i$
Analytic cond. $0.505053$
Root an. cond. $0.710671$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.50 − 0.442i)3-s + (−1.67 − 1.07i)5-s + (1.23 − 0.795i)9-s + (0.415 − 0.909i)11-s + (−3.00 − 0.881i)15-s + (−0.995 + 0.0950i)23-s + (1.23 + 2.69i)25-s + (0.485 − 0.560i)27-s + (−0.0913 − 0.0268i)31-s + (0.223 − 1.55i)33-s + (1.65 − 1.06i)37-s − 2.92·45-s + 1.68·47-s + (−0.959 + 0.281i)49-s + (0.186 + 1.29i)53-s + ⋯
L(s)  = 1  + (1.50 − 0.442i)3-s + (−1.67 − 1.07i)5-s + (1.23 − 0.795i)9-s + (0.415 − 0.909i)11-s + (−3.00 − 0.881i)15-s + (−0.995 + 0.0950i)23-s + (1.23 + 2.69i)25-s + (0.485 − 0.560i)27-s + (−0.0913 − 0.0268i)31-s + (0.223 − 1.55i)33-s + (1.65 − 1.06i)37-s − 2.92·45-s + 1.68·47-s + (−0.959 + 0.281i)49-s + (0.186 + 1.29i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261260136\)
\(L(\frac12)\) \(\approx\) \(1.261260136\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
good3 \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (1.67 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (-1.65 + 1.06i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - 1.68T + T^{2} \)
53 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.0930 + 0.647i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (-0.771 - 1.68i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.0800 - 0.0514i)T + (0.415 + 0.909i)T^{2} \)
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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

−9.504914989343959500649185020711, −8.924424688494247662969835329784, −8.218603400762763334216849591810, −7.85352982703301205834221414602, −7.04169968544184479961929796897, −5.63152213362647770483803779682, −4.17787326333452048590166744226, −3.82905902772145750874697332685, −2.70773540370956856977061150381, −1.11587726018990973564081752605, 2.28637901206095405741974425984, 3.20318012113423889100939049626, 3.96628201619784140137002621244, 4.53805613082729537231267710365, 6.45292871644619311616731007872, 7.30868052075276046987847573145, 7.921427356600204247112189786864, 8.463492677213844268304327313404, 9.544043592652214960142240997423, 10.16743748001510383523484122858

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