Properties

Label 1-1792-1792.1091-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.999 + 0.0245i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.773 − 0.634i)3-s + (−0.290 + 0.956i)5-s + (0.195 − 0.980i)9-s + (0.0980 + 0.995i)11-s + (0.956 − 0.290i)13-s + (0.382 + 0.923i)15-s + (0.382 − 0.923i)17-s + (−0.471 + 0.881i)19-s + (−0.555 − 0.831i)23-s + (−0.831 − 0.555i)25-s + (−0.471 − 0.881i)27-s + (0.995 + 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + (0.881 − 0.471i)37-s + ⋯
L(s)  = 1  + (0.773 − 0.634i)3-s + (−0.290 + 0.956i)5-s + (0.195 − 0.980i)9-s + (0.0980 + 0.995i)11-s + (0.956 − 0.290i)13-s + (0.382 + 0.923i)15-s + (0.382 − 0.923i)17-s + (−0.471 + 0.881i)19-s + (−0.555 − 0.831i)23-s + (−0.831 − 0.555i)25-s + (−0.471 − 0.881i)27-s + (0.995 + 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + (0.881 − 0.471i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 + 0.0245i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.159999060 + 0.02650850720i\)
\(L(\frac12)\) \(\approx\) \(2.159999060 + 0.02650850720i\)
\(L(1)\) \(\approx\) \(1.399517022 - 0.05593390495i\)
\(L(1)\) \(\approx\) \(1.399517022 - 0.05593390495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.773 + 0.634i)T \)
5 \( 1 + (0.290 - 0.956i)T \)
11 \( 1 + (-0.0980 - 0.995i)T \)
13 \( 1 + (-0.956 + 0.290i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (0.471 - 0.881i)T \)
23 \( 1 + (0.555 + 0.831i)T \)
29 \( 1 + (-0.995 - 0.0980i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.881 + 0.471i)T \)
41 \( 1 + (-0.831 + 0.555i)T \)
43 \( 1 + (-0.773 - 0.634i)T \)
47 \( 1 + (-0.923 - 0.382i)T \)
53 \( 1 + (-0.995 + 0.0980i)T \)
59 \( 1 + (0.956 + 0.290i)T \)
61 \( 1 + (0.634 + 0.773i)T \)
67 \( 1 + (-0.634 - 0.773i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (0.980 + 0.195i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.881 - 0.471i)T \)
89 \( 1 + (0.555 - 0.831i)T \)
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

−20.13198368111637360721035932530, −19.59390674380751988581808620401, −19.04258661517160572749973802551, −18.059445150948486614817819616850, −16.88808872971558810141444986232, −16.55524204906500315025319898669, −15.667730525226298117430133544627, −15.28142979733621538616835312930, −14.193714927643535575677912199526, −13.51574605748743913029755869439, −13.02785378893506329554609446920, −11.94864386187680974847551553295, −11.11438673539162876231167539131, −10.45821170963398798583101045700, −9.341947427716187716591744734911, −8.88586325574207705373757108493, −8.217676355258011118678594015901, −7.590144982366755583075263815122, −6.17011466059732160665393358093, −5.523850246714785967540907692217, −4.37693843139094963058627344984, −3.94284287976575170388654787461, −3.06402041749494588901082978374, −1.92163292210230363929725547544, −0.89975011434106783470617335829, 0.95990536432374412791970326777, 2.11162354252766278432177940109, 2.76346994024303262291886094806, 3.68074471088054311338970377224, 4.37294622934912935943085830400, 5.84990385342346121490635189032, 6.53641745735273953698582616658, 7.32897824409519919871034951362, 7.87094336552761982525031783350, 8.72194325685387251151962854071, 9.63403148324876808495119867135, 10.3925875363814058342300653658, 11.18736741829600540429909436326, 12.30338325840669459239072876929, 12.54680838511583430986726262375, 13.76904787060094713012920213416, 14.2680309041853859917764361799, 14.834560831096875979913407601417, 15.65118784276047210832027296913, 16.334713514102159360463232135643, 17.70611714532763605883635886634, 18.09960183945661001938724945116, 18.69651707779553500451809420846, 19.40015744008498811904469856215, 20.2142766521878777750152856365

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