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Properties

Label 1-712-712.429-r0-0-0
Degree $1$
Conductor $712$
Sign $0.999 + 0.0228i$
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  + (−0.142 − 0.989i)3-s + (−0.841 + 0.540i)5-s + (−0.841 + 0.540i)7-s + (−0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.142 − 0.989i)13-s + (0.654 + 0.755i)15-s + (−0.654 + 0.755i)17-s + (−0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (0.959 − 0.281i)23-s + (0.415 − 0.909i)25-s + (0.415 + 0.909i)27-s + (0.841 − 0.540i)29-s + (0.959 + 0.281i)31-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)3-s + (−0.841 + 0.540i)5-s + (−0.841 + 0.540i)7-s + (−0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.142 − 0.989i)13-s + (0.654 + 0.755i)15-s + (−0.654 + 0.755i)17-s + (−0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (0.959 − 0.281i)23-s + (0.415 − 0.909i)25-s + (0.415 + 0.909i)27-s + (0.841 − 0.540i)29-s + (0.959 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.999 + 0.0228i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.999 + 0.0228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6807185056 + 0.007771058373i\)
\(L(\frac12)\) \(\approx\) \(0.6807185056 + 0.007771058373i\)
\(L(1)\) \(\approx\) \(0.6849124739 - 0.1076741563i\)
\(L(1)\) \(\approx\) \(0.6849124739 - 0.1076741563i\)

Euler product

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

−22.81381931976349554851772054275, −21.66671638843303592287455261251, −20.99186369679057047305531681378, −20.13813063775397086960074077177, −19.6133499275375904199031859953, −18.68861259568932262186729424172, −17.37918644829700619583614085815, −16.729800085309891892106910654413, −15.93223422168363565032765262234, −15.52873830698518701197425886121, −14.55921094214120214671451451956, −13.41196257298461905024031148724, −12.65251496168773762293192729350, −11.60470105995562936769134271031, −10.92701195347707899440751805448, −9.94457204044199538444636760161, −9.2212988074360402950263849033, −8.387704789381348134800656220513, −7.21938738512113952876931049172, −6.38844323734557918867675613813, −4.859780402558104086745049840388, −4.55563877172647127903251871989, −3.525358662668203578438198313964, −2.506311123092653066941819470845, −0.51007944625491369463343318644, 0.74220248615461430435447047844, 2.636137361044229088791885361867, 2.86234533216879058046665210350, 4.31146040719232394869328149367, 5.74341520488248125786268461135, 6.32970180535957710309440695908, 7.27476909272017498804515683611, 8.17306241185559187532321478695, 8.72602130759078418326739221730, 10.33432846928664496720723736956, 10.91382341133809365106005872256, 11.92825691208387370210479355330, 12.7851602565809606556033821587, 13.143190609059458680154617946750, 14.392924921429148238003434900462, 15.3151051142297132042785206724, 15.87570227580426605566317058979, 17.01377793258560023021053392044, 17.88060305298897387726566559329, 18.71666140841557727660517715127, 19.2864088502202915153487180821, 19.731867048645850826583421600753, 20.96612596979291909666782183778, 22.03379071393659272430032705531, 22.84261914112835436373108791163

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