Properties

Label 2-1792-1792.1637-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  + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (0.290 + 0.956i)7-s + (0.956 + 0.290i)8-s + (0.881 − 0.471i)9-s + (−1.46 + 0.217i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.48 + 0.0727i)22-s + (−0.938 + 0.0924i)23-s + (0.634 + 0.773i)25-s + (0.0980 + 0.995i)28-s + (−0.509 − 0.686i)29-s + (0.881 + 0.471i)32-s + ⋯
L(s)  = 1  + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (0.290 + 0.956i)7-s + (0.956 + 0.290i)8-s + (0.881 − 0.471i)9-s + (−1.46 + 0.217i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−1.48 + 0.0727i)22-s + (−0.938 + 0.0924i)23-s + (0.634 + 0.773i)25-s + (0.0980 + 0.995i)28-s + (−0.509 − 0.686i)29-s + (0.881 + 0.471i)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: $\chi_{1792} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.870 - 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.187088680\)
\(L(\frac12)\) \(\approx\) \(2.187088680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 - 0.0980i)T \)
7 \( 1 + (-0.290 - 0.956i)T \)
good3 \( 1 + (-0.881 + 0.471i)T^{2} \)
5 \( 1 + (-0.634 - 0.773i)T^{2} \)
11 \( 1 + (1.46 - 0.217i)T + (0.956 - 0.290i)T^{2} \)
13 \( 1 + (0.773 + 0.634i)T^{2} \)
17 \( 1 + (-0.923 + 0.382i)T^{2} \)
19 \( 1 + (-0.0980 - 0.995i)T^{2} \)
23 \( 1 + (0.938 - 0.0924i)T + (0.980 - 0.195i)T^{2} \)
29 \( 1 + (0.509 + 0.686i)T + (-0.290 + 0.956i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.0970 + 1.97i)T + (-0.995 - 0.0980i)T^{2} \)
41 \( 1 + (0.195 + 0.980i)T^{2} \)
43 \( 1 + (-0.390 + 1.55i)T + (-0.881 - 0.471i)T^{2} \)
47 \( 1 + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (1.02 - 1.37i)T + (-0.290 - 0.956i)T^{2} \)
59 \( 1 + (0.773 - 0.634i)T^{2} \)
61 \( 1 + (0.471 + 0.881i)T^{2} \)
67 \( 1 + (-0.416 - 0.249i)T + (0.471 + 0.881i)T^{2} \)
71 \( 1 + (0.523 - 0.979i)T + (-0.555 - 0.831i)T^{2} \)
73 \( 1 + (0.831 + 0.555i)T^{2} \)
79 \( 1 + (0.858 + 1.28i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.995 - 0.0980i)T^{2} \)
89 \( 1 + (-0.980 - 0.195i)T^{2} \)
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

−9.612721295211697890659456314552, −8.653730779342128524102701232597, −7.63041160637274278444218294948, −7.25412594105632122883921865312, −6.02044089391216490193638083623, −5.52329030748289410489117978255, −4.66947594398018657142118767155, −3.78204940657488801280691559955, −2.66487111335675428671348558560, −1.86292524611070978520708540665, 1.44886703288893351813861400783, 2.60422658405687382780388609780, 3.61242318929886344217790303223, 4.66909501747735729606668604357, 4.97247672573988628862864906290, 6.17430708375190221979768074760, 6.95226224550278214818243665432, 7.78743368627955490545643058229, 8.172500851504084808599844860128, 9.875008587743448087470272522953

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