Properties

Label 2-1792-448.181-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.881 - 0.471i$
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  + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.324 + 1.63i)11-s + (−0.707 − 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.216 + 1.08i)29-s + (−0.324 + 0.216i)37-s + (0.382 + 0.0761i)43-s + (0.707 + 0.707i)49-s + (0.382 − 1.92i)53-s + 63-s + (1.92 − 0.382i)67-s + (−1.30 − 0.541i)71-s + (−0.923 + 1.38i)77-s + (0.541 + 0.541i)79-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.324 + 1.63i)11-s + (−0.707 − 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.216 + 1.08i)29-s + (−0.324 + 0.216i)37-s + (0.382 + 0.0761i)43-s + (0.707 + 0.707i)49-s + (0.382 − 1.92i)53-s + 63-s + (1.92 − 0.382i)67-s + (−1.30 − 0.541i)71-s + (−0.923 + 1.38i)77-s + (0.541 + 0.541i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.302623369\)
\(L(\frac12)\) \(\approx\) \(1.302623369\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.324 - 1.63i)T + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.382 + 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \)
71 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
83 \( 1 + (-0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.622477013637402721813738560690, −8.724758355715335702905957732041, −7.922596827693169632550445003478, −7.16637673956321214840601145867, −6.52983736992735873510218113109, −5.24954646892691534297332933697, −4.67662079254488920165577214778, −3.85622581559046011263019762048, −2.36996828376079843955614681352, −1.56911931094038941806226569205, 1.15533648557837094571871009355, 2.34785295187339635611399767298, 3.67337314230797719050101698519, 4.37590599866515529994552605244, 5.42562856790230035897691622382, 6.06308290251504574172390975411, 7.24236195955419635357085568224, 7.899678624446398701243590814499, 8.405694126926586462038111156814, 9.453778562674986194978372594865

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