Properties

Label 1-185-185.183-r0-0-0
Degree $1$
Conductor $185$
Sign $0.921 + 0.387i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (−0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 − 0.342i)4-s + i·6-s + (−0.642 − 0.766i)7-s + (0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.984 − 0.173i)12-s + (0.939 + 0.342i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)18-s + (0.984 − 0.173i)19-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 − 0.342i)4-s + i·6-s + (−0.642 − 0.766i)7-s + (0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.984 − 0.173i)12-s + (0.939 + 0.342i)13-s + (0.866 − 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)18-s + (0.984 − 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292771010 + 0.2608155546i\)
\(L(\frac12)\) \(\approx\) \(1.292771010 + 0.2608155546i\)
\(L(1)\) \(\approx\) \(1.169767740 + 0.2797809041i\)
\(L(1)\) \(\approx\) \(1.169767740 + 0.2797809041i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.642 - 0.766i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.642 - 0.766i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−27.22344102963871176659595098043, −26.26391280530288826777890630243, −25.47599559473309929190405838785, −24.580022697468005066200479348491, −22.87095490214998200229500897510, −22.249859105394112451479489835221, −21.17731367296214493463796437436, −20.2771943778509906584216397142, −19.70517649799954751794119171241, −18.57312317502621198969656874831, −17.98592593602903029596470703338, −16.33974163314060600628681871280, −15.28462970729422828891013076726, −14.20639891643686330002176740642, −13.133275085598277437244511854340, −12.43928702879311465331725153710, −11.13353689978203604884312126835, −9.89392686242685956255308587852, −9.153110165818843425738209495379, −8.37023123653320073122102577730, −6.896876943337655717330449247249, −5.04876798968879493637354056553, −3.75723602864349514393125869740, −2.82980485842328893704212787375, −1.6599814210656529644133078572, 1.218254753931942635375160543923, 3.38027543278947848188989659489, 4.20859216040772414370913143647, 6.020997740884327613813015713584, 6.949160153843633359353097462269, 7.94096221906438147256409358389, 9.01859109272841852660842936302, 9.698885157286823108161091119199, 11.1747268473275288224826045133, 13.24550703394424693238836511688, 13.45230791938976006424558452950, 14.50290087474273664794723316375, 15.62970845659695839432062514245, 16.34612214384411381841800321475, 17.48086179579090097802479475353, 18.673115143316727919023564195855, 19.37330950784525571220915372212, 20.31959746450326100819179346317, 21.64724034987770493547097925875, 22.69995281381770806917310928171, 23.810833501573735801111889403580, 24.49083407574098021280956097953, 25.44489480503471499295297637691, 26.40338468732006463266693868652, 26.66289135239292804526741099478

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