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Properties

Label 1-1792-1792.1235-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.348 + 0.937i$
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  + (−0.999 + 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (−0.973 + 0.227i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.130 + 0.991i)17-s + (0.582 − 0.812i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (−0.995 + 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.910 + 0.412i)37-s + ⋯
L(s)  = 1  + (−0.999 + 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (−0.973 + 0.227i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (0.130 + 0.991i)17-s + (0.582 − 0.812i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (−0.995 + 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.910 + 0.412i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7257299856 + 0.5044519594i\)
\(L(\frac12)\) \(\approx\) \(0.7257299856 + 0.5044519594i\)
\(L(1)\) \(\approx\) \(0.8044382021 + 0.04653545092i\)
\(L(1)\) \(\approx\) \(0.8044382021 + 0.04653545092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.999 + 0.0327i)T \)
5 \( 1 + (0.935 - 0.352i)T \)
11 \( 1 + (-0.973 + 0.227i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (0.130 + 0.991i)T \)
19 \( 1 + (0.582 - 0.812i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (-0.956 + 0.290i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (0.910 + 0.412i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.608 + 0.793i)T \)
53 \( 1 + (0.227 + 0.973i)T \)
59 \( 1 + (-0.986 - 0.162i)T \)
61 \( 1 + (0.849 - 0.528i)T \)
67 \( 1 + (-0.0327 - 0.999i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.896 + 0.442i)T \)
79 \( 1 + (-0.991 - 0.130i)T \)
83 \( 1 + (-0.0980 + 0.995i)T \)
89 \( 1 + (-0.321 + 0.946i)T \)
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

−20.28623671263069374285343112118, −18.88157500595003202606989910506, −18.4869785530870141838008805818, −17.97899915637680512825823084145, −17.07938211435725306516007045817, −16.45938832166205150576736981079, −15.92940124968567304748366301084, −14.806574339695068542494748287718, −14.06638563052594994660989927974, −13.35108642260464675731151874135, −12.55163146942828146507729822514, −11.81588842768037865190567476271, −11.02530662336803322704714738711, −10.24856522953408341975567420346, −9.77042117967474066641207543267, −8.87830071814838751623748050749, −7.46048855571760094484028497565, −7.13304040413969948207529595167, −5.98297175112190691817888625503, −5.53928809919377253539094509435, −4.79185110752668930241246382707, −3.70371187126902017244179284632, −2.44009188718728687533460880268, −1.79903964689405334145906327832, −0.39800852837423132097456863398, 1.006113796183873008180252496586, 1.950314109552171436722833212235, 2.94395473629176733537204123136, 4.24549957928394199266628188270, 5.135005504035118544665474293918, 5.60641483367670411721886838730, 6.28578505145012667288189079461, 7.39900900604112626293706460965, 7.97862070516695032392784934592, 9.36578907945369132020816379346, 9.80401706263890025937052318422, 10.62076951528499006934344540853, 11.21044570665320165299140419699, 12.302094624053241384166954474987, 12.94543597751518145442466167281, 13.28796304677970870691960153318, 14.475407886960328067446049247717, 15.34226259309688768084923758308, 15.9819834136597420568699228162, 16.930229496989065356307993957004, 17.30364853819489919713755307730, 18.1448432456451724498867938885, 18.43930466509354546678212071357, 19.752760323448776618779589316839, 20.381960858134703635285613004609

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