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Properties

Label 1-95-95.42-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.999 + 0.00899i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.999 + 0.00899i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.999 + 0.00899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004511617439 + 1.002861613i\)
\(L(\frac12)\) \(\approx\) \(0.004511617439 + 1.002861613i\)
\(L(1)\) \(\approx\) \(0.5790451097 + 0.5806842532i\)
\(L(1)\) \(\approx\) \(0.5790451097 + 0.5806842532i\)

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

−29.44481471963544052449881874479, −28.5964863773035438955285520284, −27.334081174626757667911001870336, −26.515309534619992567553553426587, −25.14219721090810673713586563208, −24.54262816845144152637670792269, −23.099953871593924233730221540678, −21.74724551903077235927794859637, −20.693868580137179766402653581186, −19.78601808427952433942967418031, −18.625709550374549503362909887948, −18.05189674407089404905520087806, −17.001510059618732160563307104, −15.31802704875695129557814739146, −13.81946165520423939900835195593, −12.86164822727446493706441318931, −11.73586285377653058742188635184, −10.816978367950378832953922290533, −9.07966821573694378432274873321, −8.22523628946841341195790731338, −7.225997726639798631654298243675, −5.313136548293688791264859514893, −3.18421672592235122486014982343, −2.07232582203243675840656995334, −0.52420205581049618402224913024, 1.9350297364086480236775642808, 4.29530612373004599091039921237, 5.17362476318645281754135213196, 6.97912336765776764152878810057, 8.15863912147735219961873219679, 9.23529077409306251517478995010, 10.33161098068257195769336955705, 11.26899380412051010518495139536, 13.48384022109700076358254295736, 14.7342794919474375597697429866, 15.20909986515891751464857240789, 16.64330723804485730821218182812, 17.28476875542004788463559448762, 18.60834429716155797497789229475, 19.93137427456494566011652146283, 20.73921259801214763601432362274, 22.05219541915425576763875028412, 23.36816904672093338816688439444, 24.26715255660902054945712081221, 25.5689305719868921710223162882, 26.366634674767908045106784348260, 27.13291766022987826666946150977, 28.06364727862091125749594925960, 28.96605056227734059816815661654, 30.79452847099114993505776175234

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