Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $-0.709 + 0.704i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $-0.709 + 0.704i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (33, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 95,\ (0:\ ),\ -0.709 + 0.704i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4613755541 + 1.118863780i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4613755541 + 1.118863780i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8628471379 + 0.8947027792i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8628471379 + 0.8947027792i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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