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Properties

Label 1-95-95.72-r0-0-0
Degree $1$
Conductor $95$
Sign $-0.709 - 0.704i$
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  + (0.642 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(0.441178\)
Root analytic conductor: \(0.441178\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (0:\ ),\ -0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4613755541 - 1.118863780i\)
\(L(\frac12)\) \(\approx\) \(0.4613755541 - 1.118863780i\)
\(L(1)\) \(\approx\) \(0.8628471379 - 0.8947027792i\)
\(L(1)\) \(\approx\) \(0.8628471379 - 0.8947027792i\)

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

−31.242504856861151267300467541458, −29.58063091696048450932667079592, −28.415308252468844789013768933131, −27.10138776123849025065476342890, −26.57768711332397809360824173643, −25.305963390259519229109280676132, −24.15594121187379585589924828421, −23.32266574065283246233009588011, −22.08327729812060024947487020651, −21.3670653585992827678544820089, −20.61852429677635156781203762076, −18.5532318187199751904444772807, −17.42325383755805830550012112927, −16.361026485193741423788253668646, −15.54758466138180666598564630974, −14.53766436452267780312579526247, −13.50333022234825850104095018631, −11.79783501124933219577295176462, −11.09373443259902656918116993472, −9.181789455723081964598637600097, −8.27029694652662559418842753358, −6.54121470786308154771779866026, −5.30495526104295395605735922909, −4.462614262084576253186155237602, −2.93874583384424857520688300602, 1.243817846017556829317516916872, 2.582050305691738308236706259, 4.47467338510360657746672707477, 5.63197242176260525266943673540, 7.05878540124952157451459000101, 8.42143287650339736318873209063, 10.390753905127251014508943395731, 11.15510552389728883280769067516, 12.43552146535656688207250323616, 13.18632881375465876906202485129, 14.288656040568581873350453847619, 15.44168395529705695909069605216, 17.45384584050725774812556594662, 18.03983208723926646634929940953, 19.35799644647074602511840055316, 20.2757302988966591387888999459, 21.26193708313579333392934886895, 22.73753093181668350775962048681, 23.32311405771839304759916557859, 24.28197394004864847068766262523, 25.263370127442951427274162590259, 26.980736443128336286914839526645, 28.19743456067697092354124708846, 28.8395040291015925008050149180, 30.17489255578163552102347790865

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