Properties

Label 2-1792-1792.517-c0-0-0
Degree $2$
Conductor $1792$
Sign $-0.870 + 0.492i$
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  + (0.773 − 0.634i)2-s + (0.195 − 0.980i)4-s + (−0.881 − 0.471i)7-s + (−0.471 − 0.881i)8-s + (−0.956 − 0.290i)9-s + (−0.439 − 0.733i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.805 − 0.288i)22-s + (−0.448 − 0.368i)23-s + (0.995 + 0.0980i)25-s + (−0.634 + 0.773i)28-s + (0.0951 − 0.0238i)29-s + (−0.956 + 0.290i)32-s + ⋯
L(s)  = 1  + (0.773 − 0.634i)2-s + (0.195 − 0.980i)4-s + (−0.881 − 0.471i)7-s + (−0.471 − 0.881i)8-s + (−0.956 − 0.290i)9-s + (−0.439 − 0.733i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.805 − 0.288i)22-s + (−0.448 − 0.368i)23-s + (0.995 + 0.0980i)25-s + (−0.634 + 0.773i)28-s + (0.0951 − 0.0238i)29-s + (−0.956 + 0.290i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.183517500\)
\(L(\frac12)\) \(\approx\) \(1.183517500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.773 + 0.634i)T \)
7 \( 1 + (0.881 + 0.471i)T \)
good3 \( 1 + (0.956 + 0.290i)T^{2} \)
5 \( 1 + (-0.995 - 0.0980i)T^{2} \)
11 \( 1 + (0.439 + 0.733i)T + (-0.471 + 0.881i)T^{2} \)
13 \( 1 + (0.0980 + 0.995i)T^{2} \)
17 \( 1 + (0.923 - 0.382i)T^{2} \)
19 \( 1 + (0.634 - 0.773i)T^{2} \)
23 \( 1 + (0.448 + 0.368i)T + (0.195 + 0.980i)T^{2} \)
29 \( 1 + (-0.0951 + 0.0238i)T + (0.881 - 0.471i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.346 + 0.968i)T + (-0.773 + 0.634i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (-0.480 + 0.0713i)T + (0.956 - 0.290i)T^{2} \)
47 \( 1 + (0.382 + 0.923i)T^{2} \)
53 \( 1 + (-1.55 - 0.390i)T + (0.881 + 0.471i)T^{2} \)
59 \( 1 + (0.0980 - 0.995i)T^{2} \)
61 \( 1 + (0.290 - 0.956i)T^{2} \)
67 \( 1 + (-1.58 + 1.17i)T + (0.290 - 0.956i)T^{2} \)
71 \( 1 + (0.482 + 1.59i)T + (-0.831 + 0.555i)T^{2} \)
73 \( 1 + (-0.555 + 0.831i)T^{2} \)
79 \( 1 + (0.162 - 0.108i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.773 + 0.634i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
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

−9.229176945868304617666447163171, −8.592200578802267824365891686951, −7.38700148645444913492413823873, −6.44900102863036765698921947069, −5.87936644159360197327400126334, −5.05251864561019911832005091811, −3.90447242821342077199283816345, −3.21141281091723341269585569975, −2.41175646759658421325313029693, −0.64936663439373444718682567722, 2.36262539271789127577855083848, 3.02701533820577620729569808845, 4.06619994780061033184929063751, 5.15353143501669307552805293841, 5.67071156508148537512672961589, 6.58354516286080557149348379277, 7.21437952110014601601564148245, 8.223515588186714237880264973577, 8.754981464293830046902186934262, 9.697648345233776987276286139366

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