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Properties

Label 1-185-185.7-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.200 - 0.979i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (0.642 + 0.766i)2-s + (0.642 − 0.766i)3-s + (−0.173 + 0.984i)4-s + 6-s + (−0.342 − 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.642 + 0.766i)12-s + (−0.984 − 0.173i)13-s + (0.5 − 0.866i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)18-s + (−0.766 − 0.642i)19-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.642 − 0.766i)3-s + (−0.173 + 0.984i)4-s + 6-s + (−0.342 − 0.939i)7-s + (−0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.642 + 0.766i)12-s + (−0.984 − 0.173i)13-s + (0.5 − 0.866i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)18-s + (−0.766 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.200 - 0.979i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.200 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9797995095 - 1.201163531i\)
\(L(\frac12)\) \(\approx\) \(0.9797995095 - 1.201163531i\)
\(L(1)\) \(\approx\) \(1.308961728 - 0.1048610502i\)
\(L(1)\) \(\approx\) \(1.308961728 - 0.1048610502i\)

Euler product

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

−27.35991143231285835584255913397, −26.4569235506809528459007506338, −25.18313110568513106181301908654, −24.52894868043645744965360801490, −22.97975711635782088734151058875, −22.33744430878165085946173295450, −21.35778463656999212857397912599, −20.774691949867254025163933299319, −19.61347755382162266962744345098, −19.057438786476950185959952143159, −17.730508145854340941217032219509, −16.12385046663489087760622473894, −15.097548082401443913861473046401, −14.70766594556399312575807134088, −13.30907403323040671519466765621, −12.48626713482707159826531787844, −11.3290740543603655701438124859, −10.09073257664500922404431912320, −9.49435879551895382091838236845, −8.31420400265645015463027209830, −6.51834641073883496746280393858, −5.04837230702422207001762436164, −4.363340953928829503993430850291, −2.80232936601443759980097280616, −2.20286208327197564877207504257, 0.36855191191227429901588307713, 2.5324980706493468567520918680, 3.58002023097552959559961215301, 4.9250892345172724686786149946, 6.45812086588898602316542860768, 7.13930795865916150619748806958, 8.16088236865944724843589311605, 9.14812566263945200904793816530, 10.829137452598917681627410093406, 12.260959114871516673006679189383, 13.25141517210584701041307762916, 13.71280831940048807056880863259, 14.81568974469110076690380549978, 15.74721904143321098795182401096, 17.04200736120043100410348189973, 17.68412991941189674971448325311, 19.08537575477435365388323696339, 19.92964776928835917795682799179, 21.03294384414340419603372341970, 22.06041007766337764661537260247, 23.25567193973119543721074853234, 23.928144228681718457903094150090, 24.63722243360737282550394802306, 25.67157730916098639640730054135, 26.4660553091651249429622378889

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