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Properties

Label 1-712-712.75-r0-0-0
Degree $1$
Conductor $712$
Sign $0.740 - 0.671i$
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  + (−0.800 − 0.599i)3-s + (−0.540 − 0.841i)5-s + (−0.212 + 0.977i)7-s + (0.281 + 0.959i)9-s + (−0.841 − 0.540i)11-s + (−0.599 + 0.800i)13-s + (−0.0713 + 0.997i)15-s + (−0.755 − 0.654i)17-s + (0.877 + 0.479i)19-s + (0.755 − 0.654i)21-s + (0.479 − 0.877i)23-s + (−0.415 + 0.909i)25-s + (0.349 − 0.936i)27-s + (0.977 + 0.212i)29-s + (0.479 + 0.877i)31-s + ⋯
L(s)  = 1  + (−0.800 − 0.599i)3-s + (−0.540 − 0.841i)5-s + (−0.212 + 0.977i)7-s + (0.281 + 0.959i)9-s + (−0.841 − 0.540i)11-s + (−0.599 + 0.800i)13-s + (−0.0713 + 0.997i)15-s + (−0.755 − 0.654i)17-s + (0.877 + 0.479i)19-s + (0.755 − 0.654i)21-s + (0.479 − 0.877i)23-s + (−0.415 + 0.909i)25-s + (0.349 − 0.936i)27-s + (0.977 + 0.212i)29-s + (0.479 + 0.877i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6743833719 - 0.2603109233i\)
\(L(\frac12)\) \(\approx\) \(0.6743833719 - 0.2603109233i\)
\(L(1)\) \(\approx\) \(0.6701647575 - 0.1422398631i\)
\(L(1)\) \(\approx\) \(0.6701647575 - 0.1422398631i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.800 - 0.599i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (-0.212 + 0.977i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.599 + 0.800i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.877 + 0.479i)T \)
23 \( 1 + (0.479 - 0.877i)T \)
29 \( 1 + (0.977 + 0.212i)T \)
31 \( 1 + (0.479 + 0.877i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.599 - 0.800i)T \)
43 \( 1 + (-0.977 + 0.212i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.800 - 0.599i)T \)
61 \( 1 + (0.936 + 0.349i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.540 - 0.841i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.0713 - 0.997i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−22.60261495840223153870583046910, −22.15446950755700277452773616302, −21.18437230667589279760102343281, −20.159713950944498086009906532508, −19.654838773077155376535891075956, −18.43110193599596888725230307030, −17.65314780852128857735542184655, −17.15578898066366222614280020983, −15.9488576770590889272461880592, −15.44161833879653409367311745704, −14.76007622999727322157153156130, −13.527189326224206984177679567348, −12.71322777520370877740535444756, −11.61258271622381039427349669827, −10.963166098036818238163457362268, −10.18217805400810537781812720138, −9.708887216885754784062788290522, −8.07036296813664446331491950727, −7.24822203491513453238241069149, −6.557364694795602446465460371049, −5.366656669351823794794052903, −4.47819623536831516819569436634, −3.600126060611724151631875067819, −2.61632012013823148506059538591, −0.71184998223925019840849318981, 0.632679599499116183344876593602, 1.980188054801019893079047142625, 3.04798916591021081666619111260, 4.77552964649559953563929701530, 5.06812207006044427505545474154, 6.2025892931313540726939528533, 7.092876726090942868377580431040, 8.13879232783494479474426361776, 8.823767684904133891666214794465, 9.91748638662339484869203780825, 11.09551346763179792524970989895, 11.898000578807748617600175912, 12.334564483239744657108831235309, 13.20133103239962037744453589670, 14.05639078032371780004867814909, 15.457116646228829146393649193534, 16.089443552419140062533155668624, 16.63471014622853219942642201829, 17.67494601755419129517365435872, 18.55506918598570639930308922225, 19.020381195253582653746649165148, 19.965705212197508886808713364665, 20.96387350565345483910581753158, 21.82622671855009795860553733426, 22.512148450156612918894211635398

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