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Properties

Label 1-183-183.20-r1-0-0
Degree $1$
Conductor $183$
Sign $-0.629 + 0.776i$
Analytic cond. $19.6660$
Root an. cond. $19.6660$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)10-s − 11-s + 13-s − 14-s + (−0.809 − 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)20-s + (−0.809 + 0.587i)22-s + (−0.309 + 0.951i)23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)10-s − 11-s + 13-s − 14-s + (−0.809 − 0.587i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)20-s + (−0.809 + 0.587i)22-s + (−0.309 + 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $-0.629 + 0.776i$
Analytic conductor: \(19.6660\)
Root analytic conductor: \(19.6660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{183} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 183,\ (1:\ ),\ -0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06696607487 + 0.1404532591i\)
\(L(\frac12)\) \(\approx\) \(0.06696607487 + 0.1404532591i\)
\(L(1)\) \(\approx\) \(1.003272984 - 0.2684932174i\)
\(L(1)\) \(\approx\) \(1.003272984 - 0.2684932174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−26.279559279303278117252263235399, −25.609860313126435404374724363707, −24.654673872931635280860210412396, −23.80058996693468052955528588431, −23.00665371180740697870332011216, −22.0434675582669769434869039331, −20.9162461000292956926691591177, −20.33533062568339469186886025649, −18.905172126497767457071894238364, −17.76472405316732552807635576983, −16.38514214761547363173768719031, −15.97526431049469758195989860519, −15.12185006129905396733915374249, −13.57755290959579295240597922417, −12.93758151771453874139707475856, −12.13918834691668486614787154849, −10.89169076582420334146180766608, −9.099319208978978297311578065540, −8.37130271853646539642643848007, −7.064005000833533693273732263393, −5.85128257042231849527846342146, −4.94621437192080327017802295064, −3.72692381482490433595998267849, −2.41196449305825116980695238565, −0.03800508259254691905396844893, 1.95474796954939466570093683506, 3.36369087144014464732413916998, 3.981441111108297960619524300888, 5.7605740394798762749826927508, 6.58486576098623224762041020156, 7.84722342843928384280623386744, 9.67378460051396732288070044307, 10.67344622558569758632407302288, 11.18250494036030353943820833316, 12.75500260550069394187207741246, 13.33787494875047243350222212566, 14.49126007315306328901830378140, 15.427403472513672983004938377304, 16.274209404330942195620896635254, 17.97408898856098852318429365765, 18.97065902354997820649857299257, 19.61236221605502100709655379431, 20.77187381770921110300711083289, 21.64675511467032879491917740110, 22.732905010135431420237576040962, 23.26868819031712604066282056818, 24.04104240239546408167877775542, 25.7201871912432230515342320325, 26.16870695633195981694047673545, 27.56043778206023147509867005254

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