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Properties

Label 1-95-95.93-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: $\chi_{95} (93, \cdot )$
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

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

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