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Properties

Label 1-1792-1792.467-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.996 + 0.0857i$
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  + (0.528 + 0.849i)3-s + (−0.162 + 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.793 − 0.608i)17-s + (0.412 + 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (−0.995 + 0.0980i)27-s + (−0.956 + 0.290i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.812 + 0.582i)37-s + ⋯
L(s)  = 1  + (0.528 + 0.849i)3-s + (−0.162 + 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.793 − 0.608i)17-s + (0.412 + 0.910i)19-s + (−0.321 − 0.946i)23-s + (−0.946 − 0.321i)25-s + (−0.995 + 0.0980i)27-s + (−0.956 + 0.290i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.812 + 0.582i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.996 + 0.0857i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.996 + 0.0857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05874528445 + 1.367807530i\)
\(L(\frac12)\) \(\approx\) \(0.05874528445 + 1.367807530i\)
\(L(1)\) \(\approx\) \(0.9104861093 + 0.6539607780i\)
\(L(1)\) \(\approx\) \(0.9104861093 + 0.6539607780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.528 + 0.849i)T \)
5 \( 1 + (-0.162 + 0.986i)T \)
11 \( 1 + (0.683 + 0.729i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (0.412 + 0.910i)T \)
23 \( 1 + (-0.321 - 0.946i)T \)
29 \( 1 + (-0.956 + 0.290i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (-0.812 + 0.582i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (-0.991 + 0.130i)T \)
53 \( 1 + (0.729 - 0.683i)T \)
59 \( 1 + (0.352 + 0.935i)T \)
61 \( 1 + (0.0327 + 0.999i)T \)
67 \( 1 + (-0.849 + 0.528i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (-0.0654 - 0.997i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (-0.0980 + 0.995i)T \)
89 \( 1 + (-0.659 - 0.751i)T \)
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

−19.68161684160148398897478148287, −19.23911893120198231021999228755, −18.646853100991859727719623449239, −17.33152380299846961860531880060, −17.19948028710237524601988335292, −16.2431787309830680522402112665, −15.35872804537296737935375090954, −14.52713738382720367715338086083, −13.73008957499815373106584209678, −13.302426248039943172266114569059, −12.26207649233477015166673972823, −11.917814716810155473532811912943, −11.10676460913309868871504523073, −9.60906047510714846130403710157, −9.220069973999821295974226333815, −8.4397789183414680899445759358, −7.6774583426541590700312363484, −6.99166344791349654085346368404, −5.95881198725587720092557429029, −5.2857236684003103464277738920, −4.03283316344739582143794652415, −3.46710885526294858889719938137, −2.167687047889245761188409756748, −1.46340219236847839222766562177, −0.43641236993118637651984600240, 1.57644248000481843854094870453, 2.719139268082469721676533992906, 3.24639769794601100073919434653, 4.11339152910009426835436567812, 4.99057971698791674950436073881, 5.869509213276625954841896125990, 6.96564463370685797084555218674, 7.6383229985688551446429080710, 8.39013331643189246443976913936, 9.47654163484655604213538444117, 10.09496424357649900051179866315, 10.48424720969862417889873942608, 11.59879996758679449589001756163, 12.17350162544760812007505591176, 13.28065909665109454937299342631, 14.36753698880352420939393852792, 14.578036417965368596203968317193, 15.159129975371496616594777897061, 16.15082750879333338332099324482, 16.67439837802081554939666720485, 17.72440124108390463378755534010, 18.34244824371455946412111618275, 19.28769672750405181080106427567, 19.8156133268356204868393137744, 20.59300377575086037670154046208

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