Properties

Degree 1
Conductor $ 2^{5} \cdot 3 $
Sign $0.980 + 0.195i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 + 0.707i)5-s i·7-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 17-s + (0.707 − 0.707i)19-s i·23-s + i·25-s + (−0.707 + 0.707i)29-s − 31-s + (0.707 − 0.707i)35-s + (−0.707 − 0.707i)37-s i·41-s + (−0.707 − 0.707i)43-s − 47-s + ⋯
L(s,χ)  = 1  + (0.707 + 0.707i)5-s i·7-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s + 17-s + (0.707 − 0.707i)19-s i·23-s + i·25-s + (−0.707 + 0.707i)29-s − 31-s + (0.707 − 0.707i)35-s + (−0.707 − 0.707i)37-s i·41-s + (−0.707 − 0.707i)43-s − 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.980 + 0.195i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.980 + 0.195i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(96\)    =    \(2^{5} \cdot 3\)
\( \varepsilon \)  =  $0.980 + 0.195i$
motivic weight  =  \(0\)
character  :  $\chi_{96} (11, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 96,\ (0:\ ),\ 0.980 + 0.195i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.117616561 + 0.1100756235i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.117616561 + 0.1100756235i\)
\(L(\chi,1)\)  \(\approx\)  \(1.138631796 + 0.06680512052i\)
\(L(1,\chi)\)  \(\approx\)  \(1.138631796 + 0.06680512052i\)

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

−29.84662351331266053787513212172, −29.18130078359413903839354161345, −27.975849300835497465303875644243, −27.274511527008346189102687708534, −25.65684948220388857756855656722, −24.89606165596556370249748353700, −24.1752491934074109508874428985, −22.54398835189259766050814628745, −21.68246498545756267689684972884, −20.75134620039302339370509910609, −19.500915750371796585984717923340, −18.37261578134300688479130772828, −17.21962152496376751788884169787, −16.29178942942155167670300401855, −14.956517587820212016550396814439, −13.82362019196230055757850311660, −12.56950784395856917607403849732, −11.69746405627148536566572633159, −9.94129333027714006767355468197, −9.07694495809633830032267133638, −7.833998907242858696543866407191, −5.94184704449509390676092300245, −5.248974539380454181464757922050, −3.28249438713922671060226792536, −1.60032994173988810197987307549, 1.76878766018023249606969926339, 3.45502946526849939209327775905, 4.96980543493160110226903899252, 6.65741046187316691792640327064, 7.38694225223955747663397182115, 9.35083213000110330276538943844, 10.19078186624298255140941519218, 11.4048451475630036568133563981, 12.82249218669244578193094372901, 14.17919890762482032915691884767, 14.65502016906573319031760691754, 16.48828248960765380325962611803, 17.31337156662503873221692622361, 18.38982969093427422940967266688, 19.62864787998262090434875802912, 20.65191938271142455890087040275, 21.90623111030883435710074318716, 22.71718944845516010666871990546, 23.87408589292072999973842833987, 25.08434331994706615436883609258, 26.131401201291099552102709093968, 26.841220189593449204603048661123, 28.15406931336532866495675665418, 29.379687069487923211228226757186, 30.06229985923218963631244510764

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