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Properties

Label 2-920-920.59-c0-0-1
Degree $2$
Conductor $920$
Sign $0.763 + 0.645i$
Analytic cond. $0.459139$
Root an. cond. $0.677598$
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  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (1.25 + 0.368i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.186 − 0.215i)11-s + (−0.797 + 0.234i)13-s + (−0.544 − 1.19i)14-s + (−0.959 − 0.281i)16-s + (0.415 − 0.909i)18-s + (−0.239 + 1.66i)19-s + (0.841 + 0.540i)20-s − 0.284·22-s + (0.841 + 0.540i)23-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (1.25 + 0.368i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.186 − 0.215i)11-s + (−0.797 + 0.234i)13-s + (−0.544 − 1.19i)14-s + (−0.959 − 0.281i)16-s + (0.415 − 0.909i)18-s + (−0.239 + 1.66i)19-s + (0.841 + 0.540i)20-s − 0.284·22-s + (0.841 + 0.540i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.763 + 0.645i$
Analytic conductor: \(0.459139\)
Root analytic conductor: \(0.677598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :0),\ 0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8809704873\)
\(L(\frac12)\) \(\approx\) \(0.8809704873\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
11 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 + 1.91T + T^{2} \)
53 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
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 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 + 0.755i)T^{2} \)
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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

−10.15754343169070784973990162444, −9.401245232221881254461202356039, −8.532892859271193810642854690529, −8.002375515474753108884511555945, −7.19425960536360579033962605141, −5.52284049555374624338475177808, −4.85567193660565645310665112983, −3.88359375005461300654366833272, −2.14696882811165661571382941467, −1.57972680270475951383716850772, 1.38084915854672269742511538605, 2.77974200084024068998259534031, 4.49775181123005378726802108310, 5.14748747754892893251140975090, 6.56254476655167989580724920303, 6.86730972649831334899271356945, 7.75353792266008844524827204846, 8.654817527180505222442972226731, 9.602217687718448545327021529900, 10.13437328471306675047284280883

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