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Properties

Label 2-1012-253.186-c0-0-0
Degree $2$
Conductor $1012$
Sign $0.869 - 0.493i$
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  + (0.0930 + 0.647i)3-s + (−0.0913 − 0.0268i)5-s + (0.548 − 0.161i)9-s + (0.841 − 0.540i)11-s + (0.00885 − 0.0616i)15-s + (0.0475 + 0.998i)23-s + (−0.833 − 0.535i)25-s + (0.427 + 0.935i)27-s + (−0.205 + 1.43i)31-s + (0.428 + 0.494i)33-s + (1.91 − 0.560i)37-s − 0.0544·45-s − 1.91·47-s + (−0.142 − 0.989i)49-s + (−0.544 + 0.627i)53-s + ⋯
L(s)  = 1  + (0.0930 + 0.647i)3-s + (−0.0913 − 0.0268i)5-s + (0.548 − 0.161i)9-s + (0.841 − 0.540i)11-s + (0.00885 − 0.0616i)15-s + (0.0475 + 0.998i)23-s + (−0.833 − 0.535i)25-s + (0.427 + 0.935i)27-s + (−0.205 + 1.43i)31-s + (0.428 + 0.494i)33-s + (1.91 − 0.560i)37-s − 0.0544·45-s − 1.91·47-s + (−0.142 − 0.989i)49-s + (−0.544 + 0.627i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126635490\)
\(L(\frac12)\) \(\approx\) \(1.126635490\)
\(L(1)\) 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 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
good3 \( 1 + (-0.0930 - 0.647i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.205 - 1.43i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-1.91 + 0.560i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.959 - 0.281i)T^{2} \)
47 \( 1 + 1.91T + T^{2} \)
53 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (-1.65 - 1.06i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (1.49 + 0.961i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (1.38 + 0.407i)T + (0.841 + 0.540i)T^{2} \)
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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

−10.03028934926133213003353761842, −9.521986217171366497177910152614, −8.721974796015678944488643960085, −7.79087533893357374506376984970, −6.82185957083746034837141107791, −5.96522272720851657603668843422, −4.86479598965151891340036974376, −3.98178433447769296776882230814, −3.20096195254817301916387774763, −1.52067993671849624553458771564, 1.39574282273654730844048537511, 2.50840867734468697156013684900, 3.95599499007202583708613868109, 4.70809610348881194760881560996, 6.09001096952472265733239936416, 6.71911945900228286649247230776, 7.63360461433886672967640516030, 8.200848431443353286339689197230, 9.456116122663746795123507124398, 9.852617820414173677731315516635

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