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Properties

Label 1-712-712.275-r1-0-0
Degree $1$
Conductor $712$
Sign $0.584 + 0.811i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4057716530 + 0.2076348953i\)
\(L(\frac12)\) \(\approx\) \(0.4057716530 + 0.2076348953i\)
\(L(1)\) \(\approx\) \(0.6319330034 - 0.2013631759i\)
\(L(1)\) \(\approx\) \(0.6319330034 - 0.2013631759i\)

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

−22.29437458338955651173950712144, −21.90053570717635533498267566554, −20.68236187972171788448199333493, −19.86787394003864144194739048051, −18.677995709040005618600347560417, −18.204677358521717315870111085082, −17.63515556474168355646122722218, −16.3318060189566979097906197296, −15.6752117263446597218356602604, −15.2192871720083490580474416566, −14.117502541118087592404173532631, −12.98230628375225779926494688733, −11.9817831032211329331811173307, −11.55784067849611375834094867030, −10.6614451490768936364009298084, −9.73493809321434811072784429208, −9.013250028576905027565982788928, −7.501279371573277811753431188708, −6.83760707336419919037970319827, −5.94260096904366927975113241479, −5.13012736607551277901146673704, −3.92480958752268346538080665818, −3.072656555874296146473446992377, −1.727640499229903847396921546038, −0.15866698353017538835570843068, 0.858225533667645521313939006575, 1.60761866488037823832248596826, 3.74270903664881858425544257854, 4.084112703828532104529634423176, 5.461927172383773624150702367609, 6.11391699646817481988682392383, 7.098549894008296181127747918219, 8.073566417674373775170111682689, 8.92561910317997394604080401281, 10.15988654136555274581551821855, 10.878204750661971142300553245680, 11.77528740465159377553494179964, 12.44353357675919687537441195453, 13.42194966048576231251546071961, 13.88455724246531315181918124650, 15.47621071458192013187480422726, 16.319085716117675193348995008189, 16.65509088535661065389366988602, 17.4024979507678598325382903701, 18.498109426665375071987987433, 19.22074977527214391344837886014, 20.07330695526421578975487003922, 20.87197972858173661525053625761, 21.8220871922114995983185965278, 22.65615010421318963177700252366

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