Properties

Label 1-712-712.269-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.735 - 0.677i$
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  + (−0.415 − 0.909i)3-s + (0.142 + 0.989i)5-s + (−0.142 − 0.989i)7-s + (−0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.415 − 0.909i)13-s + (0.841 − 0.540i)15-s + (0.841 + 0.540i)17-s + (0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)23-s + (−0.959 + 0.281i)25-s + (0.959 + 0.281i)27-s + (0.142 + 0.989i)29-s + (−0.654 − 0.755i)31-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s + (0.142 + 0.989i)5-s + (−0.142 − 0.989i)7-s + (−0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.415 − 0.909i)13-s + (0.841 − 0.540i)15-s + (0.841 + 0.540i)17-s + (0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)23-s + (−0.959 + 0.281i)25-s + (0.959 + 0.281i)27-s + (0.142 + 0.989i)29-s + (−0.654 − 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3061214273 - 0.7838918222i\)
\(L(\frac12)\) \(\approx\) \(0.3061214273 - 0.7838918222i\)
\(L(1)\) \(\approx\) \(0.7644977835 - 0.3424993539i\)
\(L(1)\) \(\approx\) \(0.7644977835 - 0.3424993539i\)

Euler product

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

−22.78394638390826168805835434101, −22.04868718613300091649828868964, −21.20881707780474776612892062999, −20.70204533172306714364942527073, −19.89436698912370912552189047704, −18.79677791668253671956546401121, −17.8523974382016826424899943237, −17.07792098014533186611618800552, −16.18561100386453653562266726028, −15.87009536481834704571152720292, −14.703010141018072471197835721272, −14.10468802672826868236825048227, −12.52029367858402438219954217585, −12.192098638027180667327674034108, −11.45181977935000498944040514697, −9.96475953132363875282814544778, −9.599462019112869961869911788593, −8.843153492561135197896561135114, −7.79026092173350149311074516485, −6.4009680963864728447287511849, −5.522505229424141629375914877413, −4.807685893316917983164908661440, −4.03394593337118519532033655583, −2.689131287063336893705424403181, −1.462562512617850230492493343093, 0.432050564779851303834479259045, 1.67061890590116308806534929092, 2.983458360596731679828706048188, 3.67142642789551940688988158797, 5.39326877081081115009848074322, 5.971340964640255962350184735729, 7.19001038834123083453036420139, 7.40611774187555762780178521976, 8.537504040028689302352489180415, 9.99423233931317095321828170101, 10.669070654245982152779154478626, 11.36210190621124713062281471499, 12.312464256969725419121693351283, 13.3610688500842406731139106325, 13.88027264342410986172242336383, 14.627007880023406353933523559979, 15.8242172028034428971454602844, 16.80010792892633767509537187423, 17.5111043966861710313463005905, 18.146035534551030921503063946193, 19.10573214070350503994194494668, 19.57799945660395995198907995058, 20.498665387733658131179171077441, 21.874426308582232247984482282074, 22.27906226605728512650274639001

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