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Properties

Label 1-1792-1792.515-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.996 + 0.0878i$
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  + (−0.352 + 0.935i)3-s + (0.729 − 0.683i)5-s + (−0.751 − 0.659i)9-s + (−0.582 − 0.812i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.849 + 0.528i)19-s + (−0.997 + 0.0654i)23-s + (0.0654 − 0.997i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.999 + 0.0327i)37-s + ⋯
L(s)  = 1  + (−0.352 + 0.935i)3-s + (0.729 − 0.683i)5-s + (−0.751 − 0.659i)9-s + (−0.582 − 0.812i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.849 + 0.528i)19-s + (−0.997 + 0.0654i)23-s + (0.0654 − 0.997i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.999 + 0.0327i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.996 + 0.0878i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.996 + 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009483357109 - 0.2155906943i\)
\(L(\frac12)\) \(\approx\) \(0.009483357109 - 0.2155906943i\)
\(L(1)\) \(\approx\) \(0.8707855635 + 0.007917560152i\)
\(L(1)\) \(\approx\) \(0.8707855635 + 0.007917560152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.352 + 0.935i)T \)
5 \( 1 + (0.729 - 0.683i)T \)
11 \( 1 + (-0.582 - 0.812i)T \)
13 \( 1 + (-0.290 - 0.956i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (-0.849 + 0.528i)T \)
23 \( 1 + (-0.997 + 0.0654i)T \)
29 \( 1 + (-0.0980 + 0.995i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (0.999 + 0.0327i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (0.634 - 0.773i)T \)
47 \( 1 + (0.793 - 0.608i)T \)
53 \( 1 + (-0.812 + 0.582i)T \)
59 \( 1 + (-0.973 + 0.227i)T \)
61 \( 1 + (-0.162 - 0.986i)T \)
67 \( 1 + (0.935 + 0.352i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (0.321 + 0.946i)T \)
79 \( 1 + (-0.130 + 0.991i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (-0.442 - 0.896i)T \)
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

−20.354363371771949000054630264570, −19.38112549570818584077932750524, −18.887001855975372634321666634309, −18.12450564900113280291557570884, −17.63968336452203956266699145828, −16.91543833470592637568045521033, −16.169775252456142612234309442339, −14.96294282621834914256957545598, −14.389491075415176681155202596154, −13.723754176436568342368269031931, −12.88137844508575695145873220779, −12.35249209262981177668025999784, −11.39695108194953264953167529538, −10.75002239496652083678553118052, −9.86569018188614279095985027838, −9.19234032986584536721386165447, −7.87095047147802948138891253451, −7.47378190910982842210024996313, −6.48375166727258363171090819171, −6.04185682077264868174552889408, −5.09805712688206940043328667078, −4.10508406680329033200675130058, −2.67757577334696561450393188886, −2.20502912793309575931683378701, −1.34620013212032389645888297051, 0.0448138444809314868715283738, 0.90816445256817888616247884618, 2.23106234838834172807856444997, 3.21880822374835846163481552051, 4.06556837281038541947892426130, 5.06738704238401346099883947367, 5.71072247807091282368238899649, 6.06098261038688646432573075961, 7.60141583863243511714425867351, 8.38592776315773904649603484677, 9.133526331010872960092816064904, 9.93920748018782083009386639617, 10.49774815638874481490842579072, 11.164444915979403716306880413482, 12.43337866457962781399423323958, 12.64704679413533428372962275873, 13.88674543160043178162632249831, 14.40035389594196189951023326745, 15.34870400250966642862979678031, 16.10510822609550341676643093926, 16.67677568472920056427388509309, 17.225501794462626737083817617766, 18.075142357426290552942611189787, 18.75993491325080382730774034511, 20.02889444602188354886778093724

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