Properties

Degree 1
Conductor 61
Sign $-0.378 - 0.925i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.406 − 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (−0.207 − 0.978i)6-s + (0.994 + 0.104i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.207 − 0.978i)10-s + i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.743 + 0.669i)17-s + ⋯
L(s,χ)  = 1  + (0.406 − 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (−0.207 − 0.978i)6-s + (0.994 + 0.104i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.207 − 0.978i)10-s + i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (−0.743 + 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.378 - 0.925i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.378 - 0.925i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $-0.378 - 0.925i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (7, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 61,\ (1:\ ),\ -0.378 - 0.925i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.536939708 - 2.289082429i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.536939708 - 2.289082429i\)
\(L(\chi,1)\)  \(\approx\)  \(1.417020081 - 1.199564842i\)
\(L(1,\chi)\)  \(\approx\)  \(1.417020081 - 1.199564842i\)

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

−32.69586388435692861009007799760, −31.70921923761483109943905190577, −30.75489907768398848029506030819, −29.61089784698422932857867514249, −27.74202401944758579632859865139, −26.57299158227935192577800181234, −26.048977358927637206634206713479, −24.63961818045445922648604268204, −24.14881957983253976829149174851, −22.06590931123849852629482364916, −21.63834131131065890944092880692, −20.502919057467707512130935722908, −18.67052741096260339483176213176, −17.44155573243131668820252416883, −16.32441884843216366556394329325, −15.01495353289169671495376058488, −14.01047882613044723374973682214, −13.49296807424794475098140541642, −11.28529021865090868263275744054, −9.525112179225768471489774871888, −8.57819174200859028203137691161, −7.158904625889116964446562653067, −5.481363363155096883301024852182, −4.27492447017180174705587498958, −2.47719855403532062615492293512, 1.53170316391269927953248781195, 2.43496440388094088528916070684, 4.385186764255375996056803801086, 5.92951717148108015290195773752, 7.956039858166477769365811552295, 9.31955498840320541462857502974, 10.431342143966585251967190079068, 12.22195565784487685774170191620, 13.03938090904368326685687879802, 14.27529829157013021422953845450, 14.95371778776520862003955578270, 17.6838688810108916931895351783, 18.09782461056520860896348833418, 19.71606517319558049341063516365, 20.575784312245981307230066269619, 21.37404596646871393498440831749, 22.7485911871717661903076194550, 24.233942161894382784308769745330, 24.92873294296256029430117063616, 26.35688474815924714822839606954, 27.73018508148762692942626802624, 28.85438055713974941503960499551, 29.99674631983953027765839845680, 30.592759802919120161298716029404, 31.69141843121817964242923338692

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