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Properties

Label 1-712-712.131-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.557 + 0.829i$
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  + (0.755 + 0.654i)3-s + (−0.959 + 0.281i)5-s + (−0.281 − 0.959i)7-s + (0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.755 + 0.654i)13-s + (−0.909 − 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (0.415 − 0.909i)21-s + (0.989 − 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (0.281 + 0.959i)29-s + (0.989 + 0.142i)31-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.959 + 0.281i)5-s + (−0.281 − 0.959i)7-s + (0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.755 + 0.654i)13-s + (−0.909 − 0.415i)15-s + (−0.415 − 0.909i)17-s + (0.989 − 0.142i)19-s + (0.415 − 0.909i)21-s + (0.989 − 0.142i)23-s + (0.841 − 0.540i)25-s + (−0.540 + 0.841i)27-s + (0.281 + 0.959i)29-s + (0.989 + 0.142i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.557 + 0.829i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.557 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6608648005 + 1.240567984i\)
\(L(\frac12)\) \(\approx\) \(0.6608648005 + 1.240567984i\)
\(L(1)\) \(\approx\) \(1.011451079 + 0.2937554307i\)
\(L(1)\) \(\approx\) \(1.011451079 + 0.2937554307i\)

Euler product

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

−22.24533445868291360993291182518, −20.85274948620463216942276786701, −20.63786009899161871298426986884, −19.48143250254145436216393761409, −18.99840255408593556636252318639, −18.277075977310507396761133913035, −17.44416890742716161054859833124, −16.05383011189080915885198863, −15.271984177291134891204762770192, −15.14186582131380319544181532851, −13.596624582380678318190169183247, −13.020800317390776498279607956974, −12.25096918619273251626978553357, −11.53415884914728304567500855933, −10.32262524454894865409166083265, −9.214398543503214785583664694877, −8.30217396598311020705455464243, −7.97932392311880999564475849998, −6.85883528581120395914580401343, −5.8465780086829772818135616158, −4.74041547012501719613519986210, −3.42352308139428026746268790017, −2.85617222231410308406122610157, −1.58797226698260880051300728971, −0.330980331222144371660868021121, 1.05624190836148203699527150793, 2.82631901626305643060049531669, 3.34063450228884908438571719271, 4.35904608544759905536921995252, 5.06914320981884403817855211445, 6.75448789267478806877410798475, 7.445780010941848258136291357, 8.287413300517194723322375309927, 9.14780995026094488004174012790, 10.171473807948807246767725201907, 10.9256519285092227790823014373, 11.564603934511033478482384129299, 13.02772169526623798514403346260, 13.67869516665750308603523131166, 14.420808141522966879213759800016, 15.42338157439156036668321722827, 16.17334665853313780271033614295, 16.41411034975892535650415866815, 17.96952222351336593531251422302, 18.80441593496357555469040190383, 19.55095004343977151545839159907, 20.30066245569480815654793884399, 20.83924683042846217400370392401, 21.777273078132041446912952776301, 22.83855267116651129156962014387

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