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Properties

Label 2-1792-1792.741-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 + (−0.197 − 1.32i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (0.0659 + 1.34i)22-s + (−0.938 + 0.0924i)23-s + (−0.634 − 0.773i)25-s + (−0.0980 − 0.995i)28-s + (−1.45 + 1.07i)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 + (−0.197 − 1.32i)11-s + (0.195 + 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (0.0659 + 1.34i)22-s + (−0.938 + 0.0924i)23-s + (−0.634 − 0.773i)25-s + (−0.0980 − 0.995i)28-s + (−1.45 + 1.07i)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: $\chi_{1792} (741, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3004408657\)
\(L(\frac12)\) \(\approx\) \(0.3004408657\)
\(L(1)\) 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 \( 1 + (0.197 + 1.32i)T + (-0.956 + 0.290i)T^{2} \)
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 \( 1 + (1.45 - 1.07i)T + (0.290 - 0.956i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.293 - 0.0143i)T + (0.995 + 0.0980i)T^{2} \)
41 \( 1 + (0.195 + 0.980i)T^{2} \)
43 \( 1 + (1.15 + 0.289i)T + (0.881 + 0.471i)T^{2} \)
47 \( 1 + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (0.825 + 0.612i)T + (0.290 + 0.956i)T^{2} \)
59 \( 1 + (-0.773 + 0.634i)T^{2} \)
61 \( 1 + (-0.471 - 0.881i)T^{2} \)
67 \( 1 + (-0.997 + 1.66i)T + (-0.471 - 0.881i)T^{2} \)
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

−9.169855248551607821831508102478, −8.080662964295901128786472212675, −8.042240315030671137326521227018, −6.85403543646977340131726149966, −6.11991701674422070355414793534, −5.31412584575241277115666533937, −3.77408428626095517868928570395, −3.09100001531110802927426505962, −1.85987263374754182818389997371, −0.28419911710835907181332045969, 1.86904716577858244363729626728, 2.61897215446926993256354728083, 3.77513835820002843181588807502, 5.26121654589960669130484254834, 5.94157122654755173645738979722, 6.67223868754591676480840075203, 7.66011955240563789642619958002, 8.228848810870426083795421757979, 9.221696018309571845930365970811, 9.544206620511886775113552967095

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