Properties

Label 1-185-185.69-r1-0-0
Degree $1$
Conductor $185$
Sign $0.913 + 0.407i$
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 + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s − i·6-s + (0.939 − 0.342i)7-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (0.984 + 0.173i)13-s + (−0.866 − 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)18-s + (0.642 − 0.766i)19-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s − i·6-s + (0.939 − 0.342i)7-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (0.984 + 0.173i)13-s + (−0.866 − 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)18-s + (0.642 − 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.913 + 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.958687966 + 0.4172083095i\)
\(L(\frac12)\) \(\approx\) \(1.958687966 + 0.4172083095i\)
\(L(1)\) \(\approx\) \(1.201776986 + 0.02031904198i\)
\(L(1)\) \(\approx\) \(1.201776986 + 0.02031904198i\)

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 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.342 + 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
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

−26.8666230716769455271145454180, −25.82317889361799046118580879503, −25.01074965806890936163861898153, −24.26359780056312652024529295857, −23.67878386320830655654080329935, −22.2672047960144424415845204638, −20.8301858240011803966807154282, −19.995140892869513825950152062379, −18.90932274115786563393156187870, −18.21397793920571881544646625076, −17.46058131539531875895275671938, −16.06091529825616889775175477950, −15.19957283283645642776584921915, −14.07549101136800505363890121749, −13.64424636412050705229741449741, −11.91874344510220010239089077236, −10.83284291505414418494807108500, −9.31116617077101745679765792430, −8.492515221602194750844520744329, −7.8376090345471714437468453344, −6.56282613959831413751440958357, −5.5859624090139602451198284747, −3.87468281193788415053698662184, −2.04446648326612416898823018397, −0.92671824492640643469187860759, 1.374446552813228329966049832218, 2.513650439133892372782607480772, 3.97422450843946115554226349607, 4.66873617973648044499687830641, 6.97982159059426452476765038258, 8.18945138224516467394551523899, 8.889540686482349724545345465174, 9.98913341230612545703955815827, 10.894213089316601101328643362122, 11.81997682381686270131803169861, 13.300596958707045656725082264369, 14.113860414252246330210954878089, 15.36587362257812269014396651897, 16.37617125811518638369187214379, 17.60170329765711040174412738955, 18.2781699794922782485439401577, 19.81206938325057510355891887469, 20.099541470315388769023468153455, 21.1098242125535695936084742033, 21.82621999144077412832401340057, 22.949925623263572223676961559855, 24.48831349441263923441547247948, 25.448836904915858156171400973296, 26.40089465651961674279958361919, 26.97122711094692118895584478740

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