Properties

Label 1-95-95.9-r0-0-0
Degree $1$
Conductor $95$
Sign $0.934 + 0.356i$
Analytic cond. $0.441178$
Root an. cond. $0.441178$
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  + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.173 + 0.984i)22-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 18-s + (−0.939 + 0.342i)21-s + (−0.173 + 0.984i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.571692994 + 0.2899569907i\)
\(L(\frac12)\) \(\approx\) \(1.571692994 + 0.2899569907i\)
\(L(1)\) \(\approx\) \(1.578159458 + 0.1664993896i\)
\(L(1)\) \(\approx\) \(1.578159458 + 0.1664993896i\)

Euler product

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

−30.098273250519332063081516386406, −29.57925368913104635764749910676, −28.44699283325503850813464615500, −26.66076109625168621466219535293, −25.77432886185773253347660027443, −24.45993883390187835403735074497, −23.85767516458756251124195992460, −23.175044943641470856167296800851, −21.84284817638075760753100048494, −20.765224934983019215067838981795, −19.618010660093505292210271669927, −18.36651169793960011929371083784, −16.99737708059424991065365713823, −16.34021950856856501891259533399, −14.4736576849830076244645743652, −13.9054061034512150883355944988, −12.82961738240976584427396797173, −11.71619309511336784301089594127, −10.72958036012125169169724160348, −8.32858434780013019276724817197, −7.39785994433572317732510582948, −6.30175843895685207422135691777, −5.05282497212988865566950424601, −3.46136663112393954914399035571, −1.76632440623587475050293720066, 2.30929494499540452280877220539, 3.67639323412865879741081794284, 5.11564430202304506665285428883, 5.698043794023226827710570017105, 7.714465745071805576471901776856, 9.52017565619962028900128537477, 10.497172697587405053648741958646, 11.68532749793297873508842668555, 12.58695125626857444353258068218, 14.17790823292114283677645583338, 15.20978044189647898871208352133, 15.71295794635145761170139075103, 17.26393434299206033266490828115, 18.65755477301279328288338449285, 20.26840475087088801799861468145, 20.81764171262074136620427107979, 21.90724193126677750095452628855, 22.63744711921003035556577046608, 23.68009492002541015124627109655, 25.01366945189344378631744064477, 25.89989367323361328665123223969, 27.72142283242071668724777519826, 27.984666376661910173003331734941, 29.24936978158029641501085661698, 30.42335578433688237400055247932

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