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Properties

Label 1-95-95.62-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.439 - 0.898i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.439 - 0.898i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.439 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3497137382 + 0.5601976959i\)
\(L(\frac12)\) \(\approx\) \(-0.3497137382 + 0.5601976959i\)
\(L(1)\) \(\approx\) \(0.5965687012 + 0.6674330367i\)
\(L(1)\) \(\approx\) \(0.5965687012 + 0.6674330367i\)

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

−29.46334248797391558856752942881, −28.39287880792974354802535081038, −27.79362144393399565653195309370, −25.776097562100778979021556828322, −24.91526869847997030433189077588, −23.68555806739592763704933859575, −22.82286385365912383263584272293, −22.18260642291360935154385210523, −20.60286308463406481440791743138, −19.74042597768436183267266145414, −18.59213269299840669682264725015, −17.95988554663793216990438418929, −16.16937018266751020037620948853, −14.90959489505560074119572099520, −13.520406887732427875352736950902, −12.7059488447199990192992140839, −11.97666966626212887907224754885, −10.62311526621822162643543396916, −9.40843568354654126379013416789, −7.59744136371661406243118357876, −6.14767269563634193962204681315, −5.22567953447397373557343620678, −3.25264545885616451345748397835, −2.01429348739387199417122852525, −0.23710456500330123924622987430, 3.269918589202958451741776769457, 4.16116066900008687923313443228, 5.640452464303071769200658192996, 6.53891879084937925916689457559, 8.206187441113791849551139205205, 9.494898084952516155217991840307, 10.83807026633224549410073409505, 12.15208694490535158407410267678, 13.49174911312073558430473760731, 14.476300880006309499456367359349, 15.82055064072091974407518211577, 16.38373637406579384791446979359, 17.25712284835781084240895170859, 18.861104045681340629236942607263, 20.47830002353351941271124702323, 21.49014426037987623326318389887, 22.24801989763059694770963550749, 23.34630296804755204400924939421, 24.024826142751495068540501580748, 25.79761619061461641226604458162, 26.20557605588313276956830417079, 27.21763913980803072457037691183, 28.612088658561208255227119446584, 29.59961267039313672724218129057, 30.94243604674977948958265943358

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