Properties

Degree 1
Conductor 83
Sign $-0.400 + 0.916i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.338 + 0.941i)2-s + (0.953 + 0.301i)3-s + (−0.771 + 0.636i)4-s + (−0.114 + 0.993i)5-s + (0.0383 + 0.999i)6-s + (−0.927 − 0.373i)7-s + (−0.859 − 0.511i)8-s + (0.817 + 0.575i)9-s + (−0.973 + 0.227i)10-s + (−0.409 − 0.912i)11-s + (−0.927 + 0.373i)12-s + (0.988 + 0.152i)13-s + (0.0383 − 0.999i)14-s + (−0.409 + 0.912i)15-s + (0.190 − 0.981i)16-s + (0.896 − 0.443i)17-s + ⋯
L(s,χ)  = 1  + (0.338 + 0.941i)2-s + (0.953 + 0.301i)3-s + (−0.771 + 0.636i)4-s + (−0.114 + 0.993i)5-s + (0.0383 + 0.999i)6-s + (−0.927 − 0.373i)7-s + (−0.859 − 0.511i)8-s + (0.817 + 0.575i)9-s + (−0.973 + 0.227i)10-s + (−0.409 − 0.912i)11-s + (−0.927 + 0.373i)12-s + (0.988 + 0.152i)13-s + (0.0383 − 0.999i)14-s + (−0.409 + 0.912i)15-s + (0.190 − 0.981i)16-s + (0.896 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.400 + 0.916i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.400 + 0.916i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.400 + 0.916i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (49, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ -0.400 + 0.916i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7034769703 + 1.075590447i$
$L(\frac12,\chi)$  $\approx$  $0.7034769703 + 1.075590447i$
$L(\chi,1)$  $\approx$  1.014555961 + 0.8603500275i
$L(1,\chi)$  $\approx$  1.014555961 + 0.8603500275i

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

−30.69635597321293901370982147811, −29.586911206056327897207828099094, −28.43528793224966832816427216526, −27.806155488608860145257164824533, −26.1675770350849343575311891346, −25.30362435292871341592222084312, −23.96181107252800011081111838528, −23.08694028069024286882219177941, −21.58141729670586588491890686767, −20.62265082309940867054106436631, −19.896952875850469168735073881141, −18.97278058994737098889064180711, −17.84738363518847331345180168157, −15.95126192599978559472783557645, −14.91324535610540768223266252756, −13.21401857198216082596260109451, −13.032122143557924368349829009251, −11.76647812086458224722604537606, −9.8331092304528788552774840148, −9.20011576642551976554983500749, −7.87484622621011855819255299532, −5.82315863292592141317381032528, −4.21050363061981514452605450240, −2.99689835097394808026368807928, −1.46558707201242138283941363422, 3.13751188428034232436055507486, 3.757214690609935909394061632129, 5.8164580055529499355125918559, 7.0786425162828045649772521734, 8.124555839043009445614021778024, 9.438592954496326402004603793473, 10.67860870545693084874580850305, 12.746638197850272721741708539530, 13.928892472913395485323596339958, 14.45276398300127247193823757214, 15.92799523133937275249836534150, 16.338595292806748963022576570691, 18.43078594261783672160326889028, 18.9237621777247487745821931504, 20.5757128436244245190272056206, 21.75725659896271319203039414382, 22.710489630182569848215228311149, 23.71581682979417254742564804826, 25.138179921127196004014231503431, 25.924757339173854641816563409065, 26.55311250710046243489077133477, 27.44204026648957553696098569036, 29.44239538243925046808138294884, 30.50767776643041730962784938737, 31.39582847284904919696867357897

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