Properties

Degree 1
Conductor 83
Sign $0.108 + 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯
L(s,χ)  = 1  + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.108 + 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.108 + 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.108 + 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (16, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.108 + 0.994i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8399871680 + 0.7534274722i$
$L(\frac12,\chi)$  $\approx$  $0.8399871680 + 0.7534274722i$
$L(\chi,1)$  $\approx$  1.084830588 + 0.5151792589i
$L(1,\chi)$  $\approx$  1.084830588 + 0.5151792589i

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

−30.54109047042829928781114729161, −29.46617235095767548585095116585, −28.73505042521013929334814663758, −27.90241475886044119696883269466, −26.50813278749146272870503713402, −24.78767686975454684133021252068, −23.74598003168081179310172968739, −23.36033135600789325045973425138, −21.980459317378360395924569986545, −21.297946919346620363185182779911, −19.83763574280088623114078917024, −19.075491026768949882535782157273, −16.92992717412535092052918438328, −16.446925043832014013469996780008, −15.314795964494617396435046402591, −13.55135177588856720571012367277, −12.76215679274967254416311980877, −11.64288040406584098528050462216, −10.764002564676658695421133505382, −9.2365540845023234351159385853, −7.04809489648262711127535761227, −6.021380841807035724645473282596, −4.65285392671030571821777117750, −3.7010378639217365164770206012, −1.16681688387130208026942130459, 2.62640004241850237489589569044, 4.12232556093047870072922901928, 5.63150432903936505928451646650, 6.5961241346014327100004416570, 7.5854179503104658616190213661, 9.95932141382335657338106187774, 11.249776845364752603156412824139, 12.1866415841278246846260718637, 13.09369992721623849392782097956, 14.81027750558449773889006111797, 15.543415209851775233070472891014, 16.641698592511709693427372447970, 17.93535160523014940391987296786, 19.093351392803320613287914476767, 20.59799960566801378957947800194, 22.03444950504501380259370922697, 22.830715038103940050709430570971, 22.9867022990939111779064076869, 24.69155692362513986116800928237, 25.44939406149688683167704264108, 26.87771750910400876346394960440, 28.10019777699950841651682539325, 29.34681339909836591582464624553, 30.05338894261507242592175564300, 31.098083312925532143422918435284

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