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Properties

Label 1-95-95.14-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.973 + 0.226i$
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  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 18-s + (−0.766 + 0.642i)21-s + (−0.939 − 0.342i)22-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 18-s + (−0.766 + 0.642i)21-s + (−0.939 − 0.342i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.973 + 0.226i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.973 + 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1389703654 - 1.210225009i\)
\(L(\frac12)\) \(\approx\) \(-0.1389703654 - 1.210225009i\)
\(L(1)\) \(\approx\) \(0.7395184242 - 0.7453026921i\)
\(L(1)\) \(\approx\) \(0.7395184242 - 0.7453026921i\)

Euler product

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

−30.70949773406556780019452913417, −29.48561136258924824553888947022, −28.463632166578355036727090401471, −27.33436001803899829953359522805, −26.316999678197422266026540214075, −24.90536916194355773584798844941, −24.22137298154220309259793008146, −23.01289770347637400279411305954, −22.26739109838360715194091848270, −21.40390778029978155327608800360, −20.34143875698203693261936712339, −18.20231455220240152649654907383, −17.59032992498896204097331917357, −16.33553969363922055258483624366, −15.380995150782514600982829733006, −14.58768524250976023863645916315, −12.8012816738334198566573807132, −12.12909051871061650388694324587, −10.9565001386911831858963454790, −9.34159366359868140047143600480, −7.718791487211997773097565622748, −6.49066143532408988218737136516, −5.19445184869830620766661679491, −4.561009752636383692488163451, −2.52299809327649747699812006649, 0.45871592230627734473796224831, 2.0196331188798450062024212676, 4.014332268185436940969205008073, 5.139714379606151531787640659943, 6.32665875214877419594873985141, 7.66309603397825702835466726436, 9.85533255418956091578679415287, 10.97459714124394763480354246042, 11.62031400839832196607208338615, 13.00188955750607193630405925224, 13.74447091574131715552400600216, 15.153727122119439427068959661, 16.53293173750450948835928820862, 17.62203717060148695920345385817, 18.88501621630869568557635351924, 19.84231137392337633136832459503, 21.22195822181831742466440380212, 21.977121179327981245500449764740, 23.089700969671510149629591187760, 24.01141804582954467832500478800, 24.46796328992775662578415308473, 26.56137471887116896506861642664, 27.58413874795955604618692629595, 28.69370175049377947998298085402, 29.5124938813841393678833375275

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