Properties

Degree 1
Conductor $ 5 \cdot 7 $
Sign $0.908 - 0.417i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s,χ)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + i·22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.908 - 0.417i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.908 - 0.417i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(35\)    =    \(5 \cdot 7\)
\( \varepsilon \)  =  $0.908 - 0.417i$
motivic weight  =  \(0\)
character  :  $\chi_{35} (2, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 35,\ (1:\ ),\ 0.908 - 0.417i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.718111101 - 0.5939212068i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.718111101 - 0.5939212068i\)
\(L(\chi,1)\)  \(\approx\)  \(2.022687173 - 0.3439784556i\)
\(L(1,\chi)\)  \(\approx\)  \(2.022687173 - 0.3439784556i\)

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

−35.59910556888870191104038261084, −34.5491197196213397853178449292, −33.139790922880960360837633984477, −31.98429806201945131277114662520, −31.14316437994159622782128062278, −30.12097472286172158098414098320, −28.91438142656687972759492667454, −26.53942157582756157533475900835, −25.94738826735708365552515402428, −24.31453507290520542980915996304, −23.96053750711599078301222154631, −22.10785158455013824657689698595, −20.9722004665645462449841208562, −19.6913951972389108501187299340, −18.16097632199931193431851527847, −16.40798799212429862133596127345, −15.09673006095517454118607126277, −13.870216007044681308131658372130, −12.99469199329751140900192168461, −11.40305618408171642726960277184, −8.9326723563768611725471447826, −7.59617739475376023513821328415, −6.195058614860654066255917608380, −4.10981455216769926587429169531, −2.45955293864876159220403598747, 2.25482763699762490408729632759, 3.800184639831668899720305117076, 5.30810329549512608468832877963, 7.53940282731089718088345180747, 9.56053378114159156516744300472, 10.71454759968517399891237260839, 12.54940421952370673502148455321, 13.75107163835168794846892456630, 15.003608723005802107184334105923, 15.93436159450332709011586603405, 18.302937061750072741358329909129, 19.954979484248753356949264425323, 20.51981840154720851774899142914, 21.867376268165700036905272143251, 22.96795193488978953650872254987, 24.567079380133396770061055546504, 25.59616376233569921601209715488, 27.188152775860030134872747315948, 28.40474534221674878824998352695, 29.879416518472621948159014955233, 31.00485230188198038257288282362, 31.80334912458809387038562252908, 32.96917005755833550252605032144, 33.84273960730530060122891935750, 35.89377509879203932391478691188

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