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Properties

Label 2-1012-253.32-c0-0-0
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  + (1.30 + 1.50i)3-s + (−0.0671 − 0.466i)5-s + (−0.421 + 2.93i)9-s + (−0.959 − 0.281i)11-s + (0.614 − 0.709i)15-s + (0.235 + 0.971i)23-s + (0.746 − 0.219i)25-s + (−3.28 + 2.11i)27-s + (1.02 − 1.18i)31-s + (−0.827 − 1.81i)33-s + (0.252 − 1.75i)37-s + 1.39·45-s − 0.284·47-s + (−0.654 − 0.755i)49-s + (0.698 − 1.53i)53-s + ⋯
L(s)  = 1  + (1.30 + 1.50i)3-s + (−0.0671 − 0.466i)5-s + (−0.421 + 2.93i)9-s + (−0.959 − 0.281i)11-s + (0.614 − 0.709i)15-s + (0.235 + 0.971i)23-s + (0.746 − 0.219i)25-s + (−3.28 + 2.11i)27-s + (1.02 − 1.18i)31-s + (−0.827 − 1.81i)33-s + (0.252 − 1.75i)37-s + 1.39·45-s − 0.284·47-s + (−0.654 − 0.755i)49-s + (0.698 − 1.53i)53-s + ⋯

Functional equation

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

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: $\chi_{1012} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.440326045\)
\(L(\frac12)\) \(\approx\) \(1.440326045\)
\(L(1)\) 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 \( 1 + (-0.235 - 0.971i)T \)
good3 \( 1 + (-1.30 - 1.50i)T + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \)
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 \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.252 + 1.75i)T + (-0.959 - 0.281i)T^{2} \)
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 \( 1 + (-0.0395 - 0.0865i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 + 0.989i)T^{2} \)
67 \( 1 + (1.11 - 0.326i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (1.38 - 0.407i)T + (0.841 - 0.540i)T^{2} \)
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 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \)
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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

−10.15548307974124063000138849621, −9.492788568208213364433154170555, −8.749307145083577531642962106904, −8.137143037558478344384149880645, −7.40599015722245542670814692097, −5.66201327543374135818359792778, −4.92838683511229549823407589941, −4.10005498222537216958811907940, −3.17535979925257212915930105040, −2.25872984685277287638829033967, 1.38153349557450129704591824539, 2.72459772011724804432070450092, 3.05808700130831586187459189317, 4.60555546707851943081427848823, 6.12213563104652057577235501031, 6.84059459324355229360241817566, 7.49080949796771109110755815276, 8.271419698569740601182508367309, 8.813182608864355519451410371429, 9.854232200094816893205227866527

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