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Properties

Label 1-185-185.28-r1-0-0
Degree $1$
Conductor $185$
Sign $0.123 + 0.992i$
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  + (−0.342 − 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s − 6-s + (0.984 − 0.173i)7-s + (0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + (−0.939 − 0.342i)19-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s − 6-s + (0.984 − 0.173i)7-s + (0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + (−0.939 − 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.123 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07764230224 - 0.06860494265i\)
\(L(\frac12)\) \(\approx\) \(-0.07764230224 - 0.06860494265i\)
\(L(1)\) \(\approx\) \(0.5571864698 - 0.4649598910i\)
\(L(1)\) \(\approx\) \(0.5571864698 - 0.4649598910i\)

Euler product

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

−27.44505377989606637957861343253, −26.64312341308505415770697002715, −26.09420951887499598213499730852, −24.77960319759289350905819340938, −24.21727618032826405271136521370, −23.04842583702382690413121209753, −21.89603192446310984198663334522, −21.20236964448163113905483255021, −19.965052041371355587562034775236, −18.87423229316525290456522785024, −17.87365857203155987277261175257, −16.72986225655454194320714384146, −16.11349794805524063761345798225, −14.9518261849594485592761317670, −14.38631610465511823750567047829, −13.402889749249371173603900740044, −11.48492527072691791605669242713, −10.527241812055065680933652463, −9.37252113011787903264069931821, −8.5061122625855313690365346701, −7.66173626601189153833591557266, −6.070829476952714062960550613464, −4.96215037539727804920190935784, −4.138270476235421614399393211964, −2.20889559731793844957543253956, 0.03697051673331892450218130021, 1.669144028536447424141469937048, 2.40594213464595406201162837590, 3.98931185231301669687214768492, 5.332267213685186994191686999991, 7.22420489258227128220059148050, 7.99893795333699079509320912217, 8.94148607992179937945311470915, 10.33114440391537706940953944152, 11.27146834675853157344908285592, 12.46772644291451607070688138588, 12.99179471036084348245722127264, 14.21623998709630585158355337405, 15.131955355420129546410089610563, 17.168666484935711475317333510795, 17.72729366182788137398357350384, 18.44474982642291088982883399296, 19.72377965129383091205344990720, 20.16331854162402436562955930507, 21.19107143902517803989522693210, 22.298488167003427210303528726664, 23.473266404618123370702542075130, 24.205256209682418257155383681934, 25.53792130913442066714358848746, 26.1868408933931006962507669077

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