Properties

Label 2-1792-1792.1469-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.757 + 0.653i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
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  + (−0.881 + 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.995 + 0.0980i)7-s + (−0.0980 + 0.995i)8-s + (0.773 − 0.634i)9-s + (0.690 − 0.761i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (−0.249 + 0.997i)22-s + (−1.11 − 0.598i)23-s + (0.290 − 0.956i)25-s + (−0.471 + 0.881i)28-s + (−1.18 − 0.0584i)29-s + (0.773 + 0.634i)32-s + ⋯
L(s)  = 1  + (−0.881 + 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.995 + 0.0980i)7-s + (−0.0980 + 0.995i)8-s + (0.773 − 0.634i)9-s + (0.690 − 0.761i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (−0.249 + 0.997i)22-s + (−1.11 − 0.598i)23-s + (0.290 − 0.956i)25-s + (−0.471 + 0.881i)28-s + (−1.18 − 0.0584i)29-s + (0.773 + 0.634i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.757 + 0.653i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.757 + 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6684353562\)
\(L(\frac12)\) \(\approx\) \(0.6684353562\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.881 - 0.471i)T \)
7 \( 1 + (0.995 - 0.0980i)T \)
good3 \( 1 + (-0.773 + 0.634i)T^{2} \)
5 \( 1 + (-0.290 + 0.956i)T^{2} \)
11 \( 1 + (-0.690 + 0.761i)T + (-0.0980 - 0.995i)T^{2} \)
13 \( 1 + (0.956 - 0.290i)T^{2} \)
17 \( 1 + (0.382 - 0.923i)T^{2} \)
19 \( 1 + (0.471 - 0.881i)T^{2} \)
23 \( 1 + (1.11 + 0.598i)T + (0.555 + 0.831i)T^{2} \)
29 \( 1 + (1.18 + 0.0584i)T + (0.995 + 0.0980i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-1.30 + 0.326i)T + (0.881 - 0.471i)T^{2} \)
41 \( 1 + (0.831 - 0.555i)T^{2} \)
43 \( 1 + (-0.0330 + 0.0923i)T + (-0.773 - 0.634i)T^{2} \)
47 \( 1 + (-0.923 - 0.382i)T^{2} \)
53 \( 1 + (-0.854 + 0.0419i)T + (0.995 - 0.0980i)T^{2} \)
59 \( 1 + (0.956 + 0.290i)T^{2} \)
61 \( 1 + (-0.634 - 0.773i)T^{2} \)
67 \( 1 + (-0.288 + 0.609i)T + (-0.634 - 0.773i)T^{2} \)
71 \( 1 + (-0.247 + 0.301i)T + (-0.195 - 0.980i)T^{2} \)
73 \( 1 + (-0.980 - 0.195i)T^{2} \)
79 \( 1 + (0.373 + 1.87i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.881 - 0.471i)T^{2} \)
89 \( 1 + (-0.555 + 0.831i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T^{2} \)
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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

−9.324615331483747374014323384499, −8.749977844472073277488475334323, −7.82134438401865177216160418128, −6.97591421979249779094280069068, −6.25568899648079005687817178404, −5.86265755490593182806170244289, −4.39246016856384827674880773792, −3.43770488628147195144999332957, −2.17119638695024500666557284975, −0.71635883191683645333341498832, 1.41948079865956334238998529470, 2.41630847704097072546658755184, 3.63485762721097310819811212134, 4.26939670505984238820134804111, 5.68354571327936265913398230874, 6.73120425333733123524870542228, 7.27539419551318722740610071100, 7.959214767457915932477372983701, 9.043444402587169293966085871978, 9.707688482384232406541614277541

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