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Properties

Label 2-1012-253.197-c0-0-1
Degree $2$
Conductor $1012$
Sign $0.627 + 0.778i$
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  + (0.396 + 0.254i)3-s + (0.815 − 1.78i)5-s + (−0.323 − 0.707i)9-s + (−0.654 + 0.755i)11-s + (0.778 − 0.500i)15-s + (0.981 + 0.189i)23-s + (−1.87 − 2.15i)25-s + (0.119 − 0.829i)27-s + (−1.67 + 1.07i)31-s + (−0.452 + 0.132i)33-s + (0.771 + 1.68i)37-s − 1.52·45-s + 0.830·47-s + (0.841 + 0.540i)49-s + (0.273 + 0.0801i)53-s + ⋯
L(s)  = 1  + (0.396 + 0.254i)3-s + (0.815 − 1.78i)5-s + (−0.323 − 0.707i)9-s + (−0.654 + 0.755i)11-s + (0.778 − 0.500i)15-s + (0.981 + 0.189i)23-s + (−1.87 − 2.15i)25-s + (0.119 − 0.829i)27-s + (−1.67 + 1.07i)31-s + (−0.452 + 0.132i)33-s + (0.771 + 1.68i)37-s − 1.52·45-s + 0.830·47-s + (0.841 + 0.540i)49-s + (0.273 + 0.0801i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237452023\)
\(L(\frac12)\) \(\approx\) \(1.237452023\)
\(L(1)\) 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 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
good3 \( 1 + (-0.396 - 0.254i)T + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (-0.815 + 1.78i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.771 - 1.68i)T + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 - 0.830T + T^{2} \)
53 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.947 + 1.09i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.827 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
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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

−9.735373064704185434735840794708, −9.193528657108574057977946493028, −8.660045837712015223406937758199, −7.77778802742524594726490924615, −6.56574446612026512858819031056, −5.46269828793907812828066280368, −4.94879576741674394357972356086, −3.92700951902354004939366176748, −2.53009311802242955725717386434, −1.24439481171539690441374541780, 2.17924985121444152266846600397, 2.72945397602541413583678879804, 3.72633871076584319906482993210, 5.48090180862028121488398938586, 5.91843558088170700508263364898, 7.17939562059721664783308929500, 7.46505164649569677513461118002, 8.639614930209855725730843637443, 9.500598957169013315643743682552, 10.51372821586203032462564542464

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