Properties

Label 1-712-712.421-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.891 - 0.453i$
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  + (−0.936 − 0.349i)3-s + (−0.989 + 0.142i)5-s + (−0.800 − 0.599i)7-s + (0.755 + 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.349 − 0.936i)13-s + (0.977 + 0.212i)15-s + (−0.540 + 0.841i)17-s + (−0.997 + 0.0713i)19-s + (0.540 + 0.841i)21-s + (0.0713 + 0.997i)23-s + (0.959 − 0.281i)25-s + (−0.479 − 0.877i)27-s + (0.599 − 0.800i)29-s + (0.0713 − 0.997i)31-s + ⋯
L(s)  = 1  + (−0.936 − 0.349i)3-s + (−0.989 + 0.142i)5-s + (−0.800 − 0.599i)7-s + (0.755 + 0.654i)9-s + (−0.142 + 0.989i)11-s + (0.349 − 0.936i)13-s + (0.977 + 0.212i)15-s + (−0.540 + 0.841i)17-s + (−0.997 + 0.0713i)19-s + (0.540 + 0.841i)21-s + (0.0713 + 0.997i)23-s + (0.959 − 0.281i)25-s + (−0.479 − 0.877i)27-s + (0.599 − 0.800i)29-s + (0.0713 − 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.891 - 0.453i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.891 - 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03286959676 - 0.1371674546i\)
\(L(\frac12)\) \(\approx\) \(0.03286959676 - 0.1371674546i\)
\(L(1)\) \(\approx\) \(0.5449876317 - 0.03143241592i\)
\(L(1)\) \(\approx\) \(0.5449876317 - 0.03143241592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.936 - 0.349i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (-0.800 - 0.599i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.349 - 0.936i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (-0.997 + 0.0713i)T \)
23 \( 1 + (0.0713 + 0.997i)T \)
29 \( 1 + (0.599 - 0.800i)T \)
31 \( 1 + (0.0713 - 0.997i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.349 + 0.936i)T \)
43 \( 1 + (0.599 + 0.800i)T \)
47 \( 1 + (0.909 + 0.415i)T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (0.936 - 0.349i)T \)
61 \( 1 + (-0.877 + 0.479i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.989 - 0.142i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (0.977 - 0.212i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−22.78682419690946197521891550384, −21.95707326663932828322970804152, −21.36706832011558351691506966206, −20.353366917384146693767793375362, −19.22754459953613452052814859360, −18.80357363879815849726259534224, −17.93281392716816658918579033971, −16.64674400991012693074036646787, −16.20534053913363070182573024232, −15.72432440306525258138703380000, −14.730220645808417414664570727021, −13.52564664778382732660504791639, −12.44727613000114109860604297206, −12.001607000951466731249927319153, −11.02846942800120265261289159151, −10.50222843249099604506062059931, −8.99302889403139811909210313658, −8.7492592412954627118810801335, −7.12774163682175785643882536494, −6.50369079435085722751050143731, −5.55565301244170853921680295812, −4.495849978085740979579905649010, −3.74802718053478650881874835938, −2.58708666740315676211320902059, −0.77988628391782788025613492663, 0.06051911597728286679548567945, 1.10939913311968233920420964723, 2.60178799103526436411025260023, 3.97461953471079895188992206193, 4.502624480116016144277460000258, 5.89912138707854070390862502615, 6.61308682151427183657494180130, 7.52325189920286764913793867892, 8.142150575850449757144429450064, 9.68528047553434137563400755711, 10.49582899623364150836367770354, 11.14794340413422190503020626516, 12.11913383406871861215468465025, 12.900367854466844122832307925694, 13.345133653202720532082815609972, 15.0160599631547304120706319863, 15.47976375149185660104455197961, 16.36197874349708115024121917904, 17.22455953986203410367711997156, 17.83143313675737650335884316960, 18.8892268977621856436415707072, 19.55319633021532791148640182241, 20.19470941912467930886740128454, 21.35857051552395602964488490523, 22.45210903557200065938715804279

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