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Properties

Label 1-1792-1792.1349-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.909 + 0.416i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (−0.683 − 0.729i)3-s + (0.910 − 0.412i)5-s + (−0.0654 + 0.997i)9-s + (−0.849 + 0.528i)11-s + (−0.995 + 0.0980i)13-s + (−0.923 − 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.162 − 0.986i)19-s + (−0.751 + 0.659i)23-s + (0.659 − 0.751i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (0.352 − 0.935i)37-s + ⋯
L(s)  = 1  + (−0.683 − 0.729i)3-s + (0.910 − 0.412i)5-s + (−0.0654 + 0.997i)9-s + (−0.849 + 0.528i)11-s + (−0.995 + 0.0980i)13-s + (−0.923 − 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.162 − 0.986i)19-s + (−0.751 + 0.659i)23-s + (0.659 − 0.751i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (0.352 − 0.935i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.909 + 0.416i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.909 + 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04041603994 - 0.1853179758i\)
\(L(\frac12)\) \(\approx\) \(-0.04041603994 - 0.1853179758i\)
\(L(1)\) \(\approx\) \(0.7950594577 - 0.1956904784i\)
\(L(1)\) \(\approx\) \(0.7950594577 - 0.1956904784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.683 - 0.729i)T \)
5 \( 1 + (0.910 - 0.412i)T \)
11 \( 1 + (-0.849 + 0.528i)T \)
13 \( 1 + (-0.995 + 0.0980i)T \)
17 \( 1 + (-0.130 + 0.991i)T \)
19 \( 1 + (0.162 - 0.986i)T \)
23 \( 1 + (-0.751 + 0.659i)T \)
29 \( 1 + (0.471 + 0.881i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (0.352 - 0.935i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.290 - 0.956i)T \)
47 \( 1 + (0.608 - 0.793i)T \)
53 \( 1 + (0.528 + 0.849i)T \)
59 \( 1 + (0.582 - 0.812i)T \)
61 \( 1 + (-0.227 - 0.973i)T \)
67 \( 1 + (-0.729 + 0.683i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.442 + 0.896i)T \)
79 \( 1 + (-0.991 + 0.130i)T \)
83 \( 1 + (-0.634 + 0.773i)T \)
89 \( 1 + (-0.946 + 0.321i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.87179697037798108951265256214, −19.781849079173548886172611359880, −18.71388553907713998114013565429, −18.11230949563790231987005613688, −17.53760109961239560509191831068, −16.69297091279056725384202984829, −16.16319017051020392190509855586, −15.33419876504301704978207560265, −14.502152125152415566641179860713, −13.90406693809019746958362999504, −12.99448577888900674772683241135, −12.07413824887414822446397444740, −11.432256283888815476128463947152, −10.41754599966519007102059731039, −10.05772903004739849547775203257, −9.45017808651935937579530512160, −8.34785569822159602953249594971, −7.394614777910948437736284793275, −6.34685378653494143009696536407, −5.83466349404111229218159994268, −5.01136181265023545577382974838, −4.32053154640429479500523521177, −2.985958042944169089262658958501, −2.501860890242166686282282343422, −1.036682040945710920855278232531, 0.042029310618663756855197373525, 1.08138621501191230087563620604, 2.12883639831547216336577522470, 2.556595875947742946714924958849, 4.243054746016218536099378444112, 5.137097361410034926534652511258, 5.591359676376783794556873025266, 6.53908595571220911792247747776, 7.25968780386772011083985192706, 8.05276975549224313806908276394, 8.996045789355144226940532072671, 9.97835424047331996804537209069, 10.48160480123024292675886381980, 11.41721925105748483748966118714, 12.46792068311727691454317617332, 12.66884638896814318303955342024, 13.580030584033031074580520851045, 14.14105216845469251169227344849, 15.25700421637246227717326678277, 16.02737519681697424482676300619, 16.96361261834457958560405593031, 17.47706520180975202124040882259, 17.915019151591198565376487712059, 18.71482074548327818869732837006, 19.68002197055045589233826796876

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