Properties

Label 1-712-712.189-r0-0-0
Degree $1$
Conductor $712$
Sign $0.904 - 0.426i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.959 + 0.281i)3-s + (−0.415 + 0.909i)5-s + (−0.415 + 0.909i)7-s + (0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.959 + 0.281i)13-s + (0.142 − 0.989i)15-s + (−0.142 − 0.989i)17-s + (0.841 − 0.540i)19-s + (0.142 − 0.989i)21-s + (−0.841 + 0.540i)23-s + (−0.654 − 0.755i)25-s + (−0.654 + 0.755i)27-s + (0.415 − 0.909i)29-s + (−0.841 − 0.540i)31-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)3-s + (−0.415 + 0.909i)5-s + (−0.415 + 0.909i)7-s + (0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.959 + 0.281i)13-s + (0.142 − 0.989i)15-s + (−0.142 − 0.989i)17-s + (0.841 − 0.540i)19-s + (0.142 − 0.989i)21-s + (−0.841 + 0.540i)23-s + (−0.654 − 0.755i)25-s + (−0.654 + 0.755i)27-s + (0.415 − 0.909i)29-s + (−0.841 − 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.904 - 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5396970097 - 0.1207520948i\)
\(L(\frac12)\) \(\approx\) \(0.5396970097 - 0.1207520948i\)
\(L(1)\) \(\approx\) \(0.6122047505 + 0.09842787759i\)
\(L(1)\) \(\approx\) \(0.6122047505 + 0.09842787759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−22.78698978396992756765889260404, −22.04062731165205252838938008588, −21.0173410272459333760225313028, −19.90645956867748913226001666788, −19.78187828700654380244822148148, −18.40223132614471557417384387962, −17.64052253323417337569820926301, −16.881532789713909802055725427525, −16.328798308035456973403182427724, −15.54667364611351443117262431478, −14.389612834025578316289841986621, −13.24137120167759094830135353755, −12.42582642229076777270083142395, −12.263232448163552528210809130383, −10.88111821816933413923855540960, −10.21171703100992509079882918459, −9.38523348609683632276686381648, −7.87543508201216843535065551981, −7.461126119308029437356939072978, −6.40214364585317040878058802420, −5.30205450982722585775643295611, −4.594622414264799559254076924, −3.72563184801015503858955997595, −1.997143097079746280808205546913, −0.88163569032974129959094186881, 0.40018841764358147138385549432, 2.38271525076166639327220480263, 3.19351909014551003076707738704, 4.39579741554196956019044997894, 5.46821253898728635089134417481, 6.13888554093298163445893619545, 7.089695730705914402770248351112, 7.93094730517476414016208073044, 9.45046203220065237252499281111, 9.85295061400572536532484299877, 11.18939570118684470450861320350, 11.49779589904468882615969692966, 12.306673561720927272787447050435, 13.40417147651540418290245966514, 14.44033463435692045676252726849, 15.407783615297268222635401578920, 15.94565775277033717739134156791, 16.6395414733320852638582115016, 17.97287160942609644272571387956, 18.25973819526434155392686750252, 19.14860697508990991506566642887, 19.94516338089935810525665687001, 21.45194124045286630989556391828, 21.73720735118955602880005201589, 22.54875488013005333181723884569

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