Properties

Label 1-95-95.47-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.607 - 0.794i$
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  + (−0.342 − 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s i·18-s + (−0.939 + 0.342i)21-s + (0.984 + 0.173i)22-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s i·18-s + (−0.939 + 0.342i)21-s + (0.984 + 0.173i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.607 - 0.794i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.607 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3961435396 - 0.8011294528i\)
\(L(\frac12)\) \(\approx\) \(0.3961435396 - 0.8011294528i\)
\(L(1)\) \(\approx\) \(0.5890683700 - 0.3945210929i\)
\(L(1)\) \(\approx\) \(0.5890683700 - 0.3945210929i\)

Euler product

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

−30.32989949456159078151516847594, −28.81037538832826227297853119503, −28.10998875316054852542391511574, −27.2143912367589390014610745101, −26.27466316370650465323311979623, −24.938491923647602104504989894036, −23.87983165551654604088473973130, −23.40671384104009967188866608335, −21.96441645856092609921803142202, −21.20409142829954067803491750731, −19.22838673762602018676877170254, −18.1142194641427731728827050785, −17.56568018646220889698575423203, −16.22088686421423248574835819836, −15.60297936781407002466972813183, −14.26498251680225059990514957302, −12.95411714369563298620629458665, −11.35635134232380093095888858939, −10.47491777815183592426936628644, −8.90976162032769241481644008349, −7.84153379765597291161278557989, −6.24407812362279484508748229014, −5.51964848356358429568130429178, −4.20271870391401749980730996461, −1.2551611494464387555835731664, 0.61268767036228033001849567836, 2.03676811515973532171030889971, 4.15198374427186548342303362989, 5.17576074244124394578879796746, 7.0820673362097199956222131469, 8.29998688433773792469342743023, 9.98711578547684939028246243262, 10.87139215506138985700140956466, 11.75586061044636603892772331700, 12.86264639310631459955092282387, 13.96245258901518987126227695842, 15.82211829998897386061433002930, 17.09810484857048906586691838487, 17.9842614346493871074655253935, 18.5754166886028051356170033783, 20.284448394259301806542830166048, 20.90691865639316608310299125871, 22.21167941509117832383011274052, 23.07769041076546352122054413078, 24.003058676277388111658929326403, 25.59222400798121085697570822830, 26.90923278686401622145366652821, 27.73236778277941801301373784396, 28.52139947799662680202368652097, 29.47368677782848676244364944060

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