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Properties

Label 1-712-712.299-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.453 + 0.891i$
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  + (0.841 − 0.540i)3-s + (0.654 + 0.755i)5-s + (0.654 + 0.755i)7-s + (0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.841 + 0.540i)13-s + (0.959 + 0.281i)15-s + (−0.959 + 0.281i)17-s + (0.415 − 0.909i)19-s + (0.959 + 0.281i)21-s + (−0.415 + 0.909i)23-s + (−0.142 + 0.989i)25-s + (−0.142 − 0.989i)27-s + (0.654 + 0.755i)29-s + (−0.415 − 0.909i)31-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (0.654 + 0.755i)5-s + (0.654 + 0.755i)7-s + (0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.841 + 0.540i)13-s + (0.959 + 0.281i)15-s + (−0.959 + 0.281i)17-s + (0.415 − 0.909i)19-s + (0.959 + 0.281i)21-s + (−0.415 + 0.909i)23-s + (−0.142 + 0.989i)25-s + (−0.142 − 0.989i)27-s + (0.654 + 0.755i)29-s + (−0.415 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.453 + 0.891i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.453 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145667751 + 1.867447787i\)
\(L(\frac12)\) \(\approx\) \(1.145667751 + 1.867447787i\)
\(L(1)\) \(\approx\) \(1.374790308 + 0.3074359109i\)
\(L(1)\) \(\approx\) \(1.374790308 + 0.3074359109i\)

Euler product

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

−21.88079948363961385631877262011, −21.15983166538622061248711010354, −20.49885128044525076601389994376, −20.04498953028746304930963770378, −19.03481057952180673044522260675, −17.92856191271931702016841909006, −17.21472611745227633979579736046, −16.266828570987037171360291085540, −15.721383411365024642108693351753, −14.42820730598109620066660269014, −14.017051843450883726797834477023, −13.222741371434469484345146857465, −12.35981057978721381783521657132, −10.98024118452296058945393256443, −10.25291467526659499755991181883, −9.58184119105845531696415373888, −8.361411866045111743010642637722, −8.13266892049452681165685391145, −6.8674971321032879890892685779, −5.400717627429082790951282330314, −4.82860964299782336802690053933, −3.85321072084740102071077534051, −2.651805122149036588813445432338, −1.73926739823677414195474754546, −0.38385266123273925259099852092, 1.66020355333391670987682711083, 2.27410244963466346454715916076, 3.00112935182226771133189433463, 4.46679585440676572916525328390, 5.46771692601402413025572696418, 6.67270890126781523974947730924, 7.29000381941526925207980904988, 8.22170272356324301108559241080, 9.27123244746949200234610348637, 9.79620113383176039564803793326, 11.02770030338672103504174741080, 11.9076024735460710350685598793, 12.89010830757732174518626467063, 13.62401992482971907611128814929, 14.4935800906856940959059477016, 15.05271827377774249143006509547, 15.74832767107762690725944644241, 17.41736757778758114324149780413, 17.9079288122264557734691730605, 18.45228286907103472693702073600, 19.44549875665356317798351203771, 20.09429207921443461019477641267, 21.207694441612219459586760271011, 21.64468482005812025814237059192, 22.5165189035951607908899896863

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