Properties

Label 2-1792-1792.1189-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.492 + 0.870i$
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  + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.956 − 0.290i)7-s + (−0.290 + 0.956i)8-s + (0.471 + 0.881i)9-s + (0.0584 − 0.0788i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−0.0727 − 0.0659i)22-s + (0.172 + 1.75i)23-s + (0.773 − 0.634i)25-s + (−0.995 + 0.0980i)28-s + (0.0988 − 0.666i)29-s + (0.471 − 0.881i)32-s + ⋯
L(s)  = 1  + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.956 − 0.290i)7-s + (−0.290 + 0.956i)8-s + (0.471 + 0.881i)9-s + (0.0584 − 0.0788i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−0.0727 − 0.0659i)22-s + (0.172 + 1.75i)23-s + (0.773 − 0.634i)25-s + (−0.995 + 0.0980i)28-s + (0.0988 − 0.666i)29-s + (0.471 − 0.881i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243912931\)
\(L(\frac12)\) \(\approx\) \(1.243912931\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0980 + 0.995i)T \)
7 \( 1 + (-0.956 + 0.290i)T \)
good3 \( 1 + (-0.471 - 0.881i)T^{2} \)
5 \( 1 + (-0.773 + 0.634i)T^{2} \)
11 \( 1 + (-0.0584 + 0.0788i)T + (-0.290 - 0.956i)T^{2} \)
13 \( 1 + (-0.634 + 0.773i)T^{2} \)
17 \( 1 + (-0.923 + 0.382i)T^{2} \)
19 \( 1 + (0.995 - 0.0980i)T^{2} \)
23 \( 1 + (-0.172 - 1.75i)T + (-0.980 + 0.195i)T^{2} \)
29 \( 1 + (-0.0988 + 0.666i)T + (-0.956 - 0.290i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.800 - 0.882i)T + (-0.0980 + 0.995i)T^{2} \)
41 \( 1 + (-0.195 - 0.980i)T^{2} \)
43 \( 1 + (1.01 + 1.69i)T + (-0.471 + 0.881i)T^{2} \)
47 \( 1 + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (-0.0713 - 0.480i)T + (-0.956 + 0.290i)T^{2} \)
59 \( 1 + (-0.634 - 0.773i)T^{2} \)
61 \( 1 + (-0.881 + 0.471i)T^{2} \)
67 \( 1 + (0.249 + 0.997i)T + (-0.881 + 0.471i)T^{2} \)
71 \( 1 + (0.979 + 0.523i)T + (0.555 + 0.831i)T^{2} \)
73 \( 1 + (-0.831 - 0.555i)T^{2} \)
79 \( 1 + (0.704 + 1.05i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.0980 + 0.995i)T^{2} \)
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

−9.517881866770412508925434617716, −8.590604111975253151373600740470, −7.938064169122681135282191321873, −7.19576616559327135444428353976, −5.82299241659177563217834681506, −4.95847404927474110727555364298, −4.41070798455180576852066870111, −3.37227595074854146029098504693, −2.18814868588096574437376678880, −1.31902674300066055393665556366, 1.21033889256746876829880759413, 2.89629648767540922176861756842, 4.12711976368928024758376039959, 4.73981532603703571479717447829, 5.61929667954453477423742783729, 6.54304531798747134074531553248, 7.10576380716618939457273306121, 8.046991837868724973825572022007, 8.692572046715962009963757328925, 9.312876360492640052682581008289

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