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Properties

Label 1-95-95.52-r0-0-0
Degree $1$
Conductor $95$
Sign $-0.947 - 0.319i$
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  + (−0.642 − 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)6-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.342 − 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + i·18-s + (−0.766 + 0.642i)21-s + (−0.342 + 0.939i)22-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.342 − 0.939i)3-s + (−0.173 + 0.984i)4-s + (−0.939 + 0.342i)6-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + (−0.342 − 0.939i)13-s + (0.173 + 0.984i)14-s + (−0.939 − 0.342i)16-s + (0.642 + 0.766i)17-s + i·18-s + (−0.766 + 0.642i)21-s + (−0.342 + 0.939i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.947 - 0.319i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ -0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09844109826 - 0.6006086724i\)
\(L(\frac12)\) \(\approx\) \(0.09844109826 - 0.6006086724i\)
\(L(1)\) \(\approx\) \(0.4828072977 - 0.5117430602i\)
\(L(1)\) \(\approx\) \(0.4828072977 - 0.5117430602i\)

Euler product

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

−31.12242189745336943663881945502, −29.111797334717695665069041376637, −28.3397504672060549765001326549, −27.43492636499187476278965415517, −26.2833459141826407300997222680, −25.75161252748203616811999877242, −24.78561449980639516465976326854, −23.28779892816688334085687053487, −22.42114462516063634819227172042, −21.12088404150241118535056115814, −19.86366893024309898905664800614, −18.98932745863336673835784642166, −17.70555875592805975237321806781, −16.37566181825622175381299470803, −15.82582928376133605319055899653, −14.7681842321104730504097958515, −13.71228689975848059076453384590, −11.87321659399841998948081112009, −10.13329951743833925893358381172, −9.64085565919297399579854665536, −8.48294765100832985696601358312, −7.095857806544854597440239975127, −5.65475188308628297460511608462, −4.42327703320778157428014223204, −2.49583477297755389244240077143, 0.74511025449881322592963152586, 2.57173992739758702876811472830, 3.61643546467546415615303384655, 6.05542231988309087738032703557, 7.539813154119690778266476623307, 8.35710282936524052313292993286, 9.7776332765217585382387141098, 10.87559206583373394574283181581, 12.3739037003792309603730347423, 13.01371531013627027091853761304, 14.12018602200271677687027188532, 16.01859806337734718550434024691, 17.1926162308816504811908489994, 18.20593215158568090874805481583, 19.25710292121462792534153560144, 19.82030796076300675170453470060, 20.9608113821875289654274933316, 22.3030005662048703505648376989, 23.39816853216645008033222697353, 24.69857658447398320121344084967, 25.86640245598931892059791984344, 26.42763306348777751096200622179, 27.80531827893020219389296716582, 29.000402725331087728262504296251, 29.674012696938874486313948172117

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