Properties

Degree 1
Conductor $ 3 \cdot 31 $
Sign $0.275 - 0.961i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + 23-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.275 - 0.961i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.275 - 0.961i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(93\)    =    \(3 \cdot 31\)
\( \varepsilon \)  =  $0.275 - 0.961i$
motivic weight  =  \(0\)
character  :  $\chi_{93} (26, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 93,\ (0:\ ),\ 0.275 - 0.961i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5030121317 - 0.3792842022i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5030121317 - 0.3792842022i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6608640592 - 0.2177771713i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6608640592 - 0.2177771713i\)

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

−30.287571264932806443664354051483, −29.06295899409892653011743349976, −28.69407472334367866734462749187, −27.23877068788926398714354192241, −26.29757873127491111997605096385, −25.58115350219669139146037214450, −24.59740751984430180993975567885, −23.23173488044152713930132985446, −21.65557292872510078243895370270, −21.18310254014652903776790141986, −19.32157837768897284799028589491, −18.81896451762065033480214973420, −17.880503885061770256892659940534, −16.598613887815744922242168219065, −15.57826621794380813380195178714, −14.45632502907991137757927964963, −12.883101765303167289501965332546, −11.3870361628799284387019833859, −10.50863019967681817792778143093, −9.28535781844823852902698593779, −8.266963899935976682043150547144, −6.618274428041417051200791619049, −5.92015975175591033088839575449, −3.23562385854239283464509972764, −2.01590928957114098427800219244, 0.93403853861774390046758282482, 2.7151231870634652603114825055, 4.79443011506698517683412334398, 6.43799747123175904160896633382, 7.63037254767607920272146871909, 8.93860963852896750681124917406, 9.92291289593095831559468019779, 10.9245967800294312485007822129, 12.549823235784794549850915117541, 13.46348810402164979884481294707, 15.33268895298285426346852315464, 16.26546087199472152091246365806, 17.344239299923766515489948076753, 18.011004725669353561170900943522, 19.59497337936404907163671257526, 20.35160126867346347808154875825, 21.12618053564630619813976750131, 22.86990370186513397746244314317, 24.01188487639917050049370159052, 25.21342801131584562775142723204, 25.8183377411343652945992652630, 27.03335585680054120798671369748, 28.04646758895627596624349635883, 28.93524413012744737571136566799, 29.68189182523934989696797729082

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