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Properties

Label 2-1012-253.87-c0-0-0
Degree $2$
Conductor $1012$
Sign $-0.952 - 0.305i$
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  + (−0.759 + 0.876i)3-s + (−0.205 + 1.43i)5-s + (−0.0492 − 0.342i)9-s + (−0.959 + 0.281i)11-s + (−1.09 − 1.26i)15-s + (0.723 + 0.690i)23-s + (−1.05 − 0.308i)25-s + (−0.638 − 0.410i)27-s + (−1.21 − 1.40i)31-s + (0.481 − 1.05i)33-s + (−0.0135 − 0.0941i)37-s + 0.500·45-s − 0.284·47-s + (−0.654 + 0.755i)49-s + (0.698 + 1.53i)53-s + ⋯
L(s)  = 1  + (−0.759 + 0.876i)3-s + (−0.205 + 1.43i)5-s + (−0.0492 − 0.342i)9-s + (−0.959 + 0.281i)11-s + (−1.09 − 1.26i)15-s + (0.723 + 0.690i)23-s + (−1.05 − 0.308i)25-s + (−0.638 − 0.410i)27-s + (−1.21 − 1.40i)31-s + (0.481 − 1.05i)33-s + (−0.0135 − 0.0941i)37-s + 0.500·45-s − 0.284·47-s + (−0.654 + 0.755i)49-s + (0.698 + 1.53i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5786900448\)
\(L(\frac12)\) \(\approx\) \(0.5786900448\)
\(L(1)\) 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 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.723 - 0.690i)T \)
good3 \( 1 + (0.759 - 0.876i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (0.205 - 1.43i)T + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (1.21 + 1.40i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.0135 + 0.0941i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.142 + 0.989i)T^{2} \)
47 \( 1 + 0.284T + T^{2} \)
53 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-1.91 - 0.560i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-1.30 + 1.50i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.264 - 1.83i)T + (-0.959 - 0.281i)T^{2} \)
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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

−10.59986764000061405993823508945, −10.00927455470843601676821665579, −9.159763156358190984050341156056, −7.72675985769870959546241162797, −7.28459194949672361574123967325, −6.13160386405070382899119914857, −5.42077214052895369188957083906, −4.41914011477407364429656038974, −3.41034684223083079243841821355, −2.37239111456990784558102067393, 0.58905843309168270188019007433, 1.83826337412268708660688250405, 3.49367822812316129098984499408, 4.95970519464804423675940833490, 5.27608438020475588824012463911, 6.39350465223326836224972084952, 7.22994261729531215023319377855, 8.182786535205535094535808109284, 8.760523978622270764997931750150, 9.727536235043716200531643613988

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