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Properties

Label 1-812-812.571-r1-0-0
Degree $1$
Conductor $812$
Sign $0.570 + 0.820i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (−0.955 + 0.294i)3-s + (0.365 + 0.930i)5-s + (0.826 − 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)15-s + (−0.5 − 0.866i)17-s + (−0.955 − 0.294i)19-s + (0.988 − 0.149i)23-s + (−0.733 + 0.680i)25-s + (−0.623 + 0.781i)27-s + (0.988 + 0.149i)31-s + (0.955 + 0.294i)33-s + (0.826 − 0.563i)37-s + (0.733 − 0.680i)39-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)3-s + (0.365 + 0.930i)5-s + (0.826 − 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)15-s + (−0.5 − 0.866i)17-s + (−0.955 − 0.294i)19-s + (0.988 − 0.149i)23-s + (−0.733 + 0.680i)25-s + (−0.623 + 0.781i)27-s + (0.988 + 0.149i)31-s + (0.955 + 0.294i)33-s + (0.826 − 0.563i)37-s + (0.733 − 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.570 + 0.820i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.570 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9092162723 + 0.4751268936i\)
\(L(\frac12)\) \(\approx\) \(0.9092162723 + 0.4751268936i\)
\(L(1)\) \(\approx\) \(0.7271997337 + 0.1491178714i\)
\(L(1)\) \(\approx\) \(0.7271997337 + 0.1491178714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.955 + 0.294i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.955 - 0.294i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 + (0.988 + 0.149i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (-0.988 - 0.149i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (-0.0747 - 0.997i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (-0.826 + 0.563i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−21.74848430538033189881198131428, −21.3258748548100128762611142600, −20.34842064124179056788044152851, −19.46769088009864298557287291063, −18.622828194719649418933570929286, −17.444086707287169925167284202011, −17.38155686722480467662244358075, −16.4560424077111700703075097301, −15.550656519101396372878527507726, −14.77110246821903190199748392035, −13.281108022490736723007011425232, −12.89812625085174106529311138543, −12.28935339206638426321018047264, −11.25921760880541782167292923881, −10.308188227122811281168227132255, −9.72452804950961592324567668377, −8.45006310788013090336599592606, −7.690230726717482153155870790771, −6.61199952665786015252158606169, −5.758030809307033126595953907765, −4.89002101706901509067183404815, −4.35513120044298673677820735354, −2.54386587215306887242302003593, −1.59037280022670487280624709662, −0.448972607660311696345252289886, 0.58147353997754605919053171910, 2.21341537806063431998575792252, 3.054150646662338030323074807429, 4.42556552827909327922142923449, 5.160205182460617805216706552710, 6.19624931989452940844619507370, 6.83492973148814387328445917361, 7.70452440773628056333191440010, 9.14470907755305503995462636436, 9.8895106059793750021713656111, 10.8752853749813025462565722649, 11.1389458500976679891973996387, 12.25385600731041774231313158955, 13.149362184864268150343609618058, 14.04680867968636687048951502568, 15.02789043286954847149610630736, 15.66354177310871715371853029165, 16.62834738656139842991290662341, 17.35084917536862246831010279403, 18.08968243025296500728029895338, 18.789506396425741980486357220198, 19.53329991488789384129473443646, 21.01084284688300506368154126155, 21.37121778322892083408372912934, 22.21395852540296586558124683584

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