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Properties

Label 1-93-93.26-r0-0-0
Degree $1$
Conductor $93$
Sign $0.275 - 0.961i$
Analytic cond. $0.431890$
Root an. cond. $0.431890$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5030121317 - 0.3792842022i\)
\(L(\frac12)\) \(\approx\) \(0.5030121317 - 0.3792842022i\)
\(L(1)\) \(\approx\) \(0.6608640592 - 0.2177771713i\)
\(L(1)\) \(\approx\) \(0.6608640592 - 0.2177771713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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