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Properties

Label 1-95-95.88-r0-0-0
Degree $1$
Conductor $95$
Sign $0.991 - 0.127i$
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  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s − i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s − i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9191528630 - 0.05884527032i\)
\(L(\frac12)\) \(\approx\) \(0.9191528630 - 0.05884527032i\)
\(L(1)\) \(\approx\) \(0.9452109554 - 0.06355692387i\)
\(L(1)\) \(\approx\) \(0.9452109554 - 0.06355692387i\)

Euler product

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

−29.999784790118791622389068102405, −29.239006686510723176550355395463, −27.808525068609996749905408080657, −27.11678231071668760008881723454, −25.85314589376245965589644624706, −24.96570242834065587295872955955, −24.607917414055833160359005743663, −23.191186253814226835790871574621, −21.62961250893157715305055937495, −20.250114422530522082530394775336, −19.40359210175515103409499878287, −18.55868239202358882996257963073, −17.58513925508679577952902131993, −16.25958193128096865105953996142, −14.89008610455928670641290497733, −14.49111739253738591783252154434, −12.713539649326962386350214632900, −11.52789204258280998488223151785, −9.67845695439024174604692812303, −9.02653878264752100285167661727, −7.83223289911116261247481629873, −6.80383117295982513791911911156, −5.38199802318896379831780062290, −3.01352895224316356312594537840, −1.58348682469929495762951840284, 1.64524898387273256848166656262, 3.280817955322843498243209889044, 4.34338036253021673064181784620, 6.925612721765482170710184870257, 7.93085360561015145759414073561, 9.21934367625891441188778725846, 10.01392754016456614584385853345, 11.13206117608775653126856791853, 12.578807459511616275393282226606, 13.95123142725517641567263700722, 15.03658561919824872152800397615, 16.63811213004985408389921665450, 17.06079801642375459489721714729, 18.85181449382051277910892743847, 19.60280399375680085570424207946, 20.44026407351372806135985740228, 21.38171313516966552259827427514, 22.45812984064728908507401614377, 24.25721085818293273949101388628, 25.32732382418260580458770294862, 26.27184439362836933090795477601, 27.05535020422464662250435147172, 27.76103082598314813172572564066, 29.19933423769492854208036616345, 30.14478302374215218367829815622

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