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Properties

Label 1-712-712.451-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.0853 + 0.996i$
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  + (0.349 + 0.936i)3-s + (0.989 + 0.142i)5-s + (0.599 + 0.800i)7-s + (−0.755 + 0.654i)9-s + (0.142 + 0.989i)11-s + (0.936 − 0.349i)13-s + (0.212 + 0.977i)15-s + (0.540 + 0.841i)17-s + (0.0713 − 0.997i)19-s + (−0.540 + 0.841i)21-s + (−0.997 − 0.0713i)23-s + (0.959 + 0.281i)25-s + (−0.877 − 0.479i)27-s + (0.800 − 0.599i)29-s + (−0.997 + 0.0713i)31-s + ⋯
L(s)  = 1  + (0.349 + 0.936i)3-s + (0.989 + 0.142i)5-s + (0.599 + 0.800i)7-s + (−0.755 + 0.654i)9-s + (0.142 + 0.989i)11-s + (0.936 − 0.349i)13-s + (0.212 + 0.977i)15-s + (0.540 + 0.841i)17-s + (0.0713 − 0.997i)19-s + (−0.540 + 0.841i)21-s + (−0.997 − 0.0713i)23-s + (0.959 + 0.281i)25-s + (−0.877 − 0.479i)27-s + (0.800 − 0.599i)29-s + (−0.997 + 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.0853 + 0.996i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.0853 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432250450 + 1.560144118i\)
\(L(\frac12)\) \(\approx\) \(1.432250450 + 1.560144118i\)
\(L(1)\) \(\approx\) \(1.332948294 + 0.7009957565i\)
\(L(1)\) \(\approx\) \(1.332948294 + 0.7009957565i\)

Euler product

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

−22.43541891119776983874563353826, −21.365233662891431307217780636401, −20.67884133518921218271675055477, −20.12014398365418507510595831680, −18.91887971432972773524438832528, −18.33876402796799000890317206194, −17.6552530374119054592303595144, −16.75563106879736463013162717533, −16.08845410634231167837131556739, −14.34275276944512663722592061803, −14.11809544461065038202118243505, −13.531731389267548446038227922371, −12.56680075144749894336017477295, −11.58824162568098294544810017715, −10.737835552017577580957930909723, −9.702743849591816691731786661299, −8.685181186354954487055973887342, −8.036212938533657404476957260496, −7.00233060150588228300949757114, −6.120427263180373811660160311026, −5.40320718664173421725935704532, −3.93239541441134226015671560351, −2.93762273417988115405875307231, −1.64390044543604353882829668119, −1.07831422410845794095347673112, 1.71661263661385062809527262463, 2.456268145726086220884971343901, 3.6266771835176755953196099011, 4.72042920730248807770863451536, 5.52954307972324488560979976869, 6.29096449501369208196406435619, 7.74018870143527997056976170319, 8.70642052733681723764989863887, 9.3210317748172521965654875378, 10.243184171290713630826617565100, 10.8587148007250477227698267237, 11.95720132879963840836896493270, 12.97558071182837353025833060613, 13.96316052857693774635172034306, 14.64467104402316329657609932111, 15.35387459769572849255702538747, 16.0893874965460752831140425389, 17.291842285140245705272484471768, 17.754834076530720348457288926087, 18.630503534441451994817956850242, 19.81154329663975230342510067608, 20.55979635834032902185275184532, 21.29993794355095288904278670368, 21.79386296215049141962770945575, 22.53439813278386077223200248628

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