Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.939 - 0.341i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 18-s + (−0.173 − 0.984i)21-s + (0.766 − 0.642i)22-s + ⋯
L(s,χ)  = 1  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 18-s + (−0.173 − 0.984i)21-s + (0.766 − 0.642i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.939 - 0.341i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.939 - 0.341i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.939 - 0.341i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (79, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (1:\ ),\ 0.939 - 0.341i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.099023001 - 0.3694081111i$
$L(\frac12,\chi)$  $\approx$  $2.099023001 - 0.3694081111i$
$L(\chi,1)$  $\approx$  1.421858729 + 0.07750006576i
$L(1,\chi)$  $\approx$  1.421858729 + 0.07750006576i

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

−30.541182797504876964040066966998, −28.70354483705808549040918192103, −28.01565728182748669020159934548, −27.14852620911129502305723306102, −25.97200827657479229387217679715, −24.95158505286936344116618621045, −23.4339731370685970355333315390, −22.291743288070841142520196965430, −21.25293886883521270073841680439, −20.69463610756062607529241886508, −19.579615082551631265369458894182, −18.55208424007533454130299909846, −17.44651211523117486839891275732, −15.46685897399951448647871510857, −14.8953541901138806899684492633, −13.53389267626831317783201045788, −12.53546900750803503576565365498, −11.099150073308508716257034213498, −10.13368708091746674072581395485, −8.94932532310373905004733394107, −8.05265677472669380825734932288, −5.53261961734548365616270152572, −4.3925302826580647141317653687, −3.02004940579589628025304414641, −1.83359593798650760732477226205, 0.911379565306398928369655305141, 3.18027835745301044818227633007, 4.5764281560229992446386055533, 6.28265109558496556166590415692, 7.37012175533149548442046968713, 8.29909850916286147682538216432, 9.36622231922939844268857520005, 11.21973086763165950301861306334, 12.96381541936076682901105833801, 13.75787750010725685825650251115, 14.474380005648325407019843357210, 15.81956902195692515379194373387, 16.90771705222741417844927226795, 18.18703861297390185530206286029, 18.89341701603248496858016829551, 20.427325668844907664164168228932, 21.37142635254126394704349836757, 23.01168598929626473220810600586, 23.84736796551711123793995926542, 24.544878077546222347458524030248, 25.67918837245098955647092305039, 26.53827775388278389424962944709, 27.259845066593016378359594451465, 29.04037156115503409330219332046, 30.19847810436809557943029428003

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