Properties

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

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 + 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(96\)    =    \(2^{5} \cdot 3\)
\( \varepsilon \)  =  $-0.195 + 0.980i$
motivic weight  =  \(0\)
character  :  $\chi_{96} (53, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 96,\ (1:\ ),\ -0.195 + 0.980i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9648175001 + 1.175633525i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9648175001 + 1.175633525i\)
\(L(\chi,1)\)  \(\approx\)  \(1.037047072 + 0.3977862730i\)
\(L(1,\chi)\)  \(\approx\)  \(1.037047072 + 0.3977862730i\)

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

−29.631026959405783107467904113585, −28.58295342044434504544986683121, −27.64497990067759501677561052631, −26.35608625473330342167845775280, −25.47922975971426750393724210909, −24.35877238835662353464923587412, −23.38919209493296263007175305092, −22.27182291005911326485834773427, −20.827970234844865582547928388233, −20.40346420885468122508773012006, −19.00507172188109317200681969160, −17.45663095646813714451077180289, −17.04748723300061775558249053502, −15.631440990995737388369720108872, −14.29078769588238132703768325045, −13.173451142504524051990278116079, −12.34622489996258700446846059346, −10.505241523279143754695982066956, −9.820212179307148950508770476667, −8.27032142840661838681767410892, −7.065212666372232272672515731970, −5.4643497165012661788790223054, −4.36427024933083804245328397739, −2.42359928932113692134862530247, −0.66836824823783880605269360371, 1.98806698935310155849984104960, 3.21401345039426940087606737065, 5.28877304309692124524815057806, 6.2221296589816990829602890439, 7.72791828215196771422872318732, 9.16522178574578139933632837627, 10.237531056184387963008915056555, 11.50039242745869136867271677604, 12.73206473066072292821565355031, 14.06179460747018106452466748357, 14.919633056726641169313050084393, 16.21657870661996581507932456051, 17.456797528858309709581569984632, 18.64262313991719246779269057982, 19.17399034747747946844149848887, 21.166752131518055774254678849810, 21.54157193056800171129809062105, 22.71485409274994313959488474853, 23.97332808585899609039542242040, 25.12316221417358690285077896122, 25.92831490680604368435959265660, 26.98513172616604793702112885686, 28.203449527312524821927095976575, 29.28069630917444538882474840806, 29.95041386620259439173626883496

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