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Properties

Label 1-712-712.435-r1-0-0
Degree $1$
Conductor $712$
Sign $0.144 - 0.989i$
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  + (−0.989 + 0.142i)3-s + (0.841 − 0.540i)5-s + (0.540 + 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.989 + 0.142i)13-s + (−0.755 + 0.654i)15-s + (0.654 − 0.755i)17-s + (−0.281 − 0.959i)19-s + (−0.654 − 0.755i)21-s + (−0.281 − 0.959i)23-s + (0.415 − 0.909i)25-s + (−0.909 + 0.415i)27-s + (−0.540 − 0.841i)29-s + (−0.281 + 0.959i)31-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)3-s + (0.841 − 0.540i)5-s + (0.540 + 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.989 + 0.142i)13-s + (−0.755 + 0.654i)15-s + (0.654 − 0.755i)17-s + (−0.281 − 0.959i)19-s + (−0.654 − 0.755i)21-s + (−0.281 − 0.959i)23-s + (0.415 − 0.909i)25-s + (−0.909 + 0.415i)27-s + (−0.540 − 0.841i)29-s + (−0.281 + 0.959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129711439 - 0.9769358257i\)
\(L(\frac12)\) \(\approx\) \(1.129711439 - 0.9769358257i\)
\(L(1)\) \(\approx\) \(0.9617673838 - 0.1102319088i\)
\(L(1)\) \(\approx\) \(0.9617673838 - 0.1102319088i\)

Euler product

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

−22.56062578089818671032093432875, −21.75485323424757042578704948518, −21.32415038842126830356783190199, −20.16125815239750174525330010613, −19.13733057634644543853179589988, −18.42383512283679412193525995516, −17.41899407407746511203771986193, −17.057692141937810692745138130354, −16.46350679039587919839587470444, −14.96963364932677840740706301085, −14.36108720289885552417380424014, −13.503662843521498053728509793919, −12.57294491639180610995797679184, −11.64942736188999653393364648606, −10.84095081579775586953129232922, −10.176036592394655550219866534329, −9.420211527139878147348841144515, −7.85861944966028665737757059537, −7.18215408420322856587412668940, −6.14275560449184426898834333261, −5.596980984657680226578021475307, −4.44347849835647196636339400576, −3.44858974235722345669367601402, −1.83525085013077083515104938060, −1.13864558130638451201279891918, 0.417276192150597174126118725446, 1.6285061533589318843526884503, 2.52384532326275365708460743237, 4.369950403994835225236489866937, 4.956257462856063263054518187635, 5.75139008320881430430361053363, 6.61936553949247254114423934062, 7.61230033453288697010267659402, 9.09779380344313096995261472460, 9.44837186013171655030933598935, 10.48743790369961940159954811806, 11.50450182937203273189687648590, 12.29540637080879207566941768693, 12.69981902424017632370442734179, 14.08468324240284217313203266601, 14.766737457191431497231452282548, 15.81283469511580490034390878755, 16.653587618157084815005440783504, 17.41557264601859672469972388174, 17.84024631798485529066968186671, 18.75107207580574976127735020967, 19.825375785371841534233594012173, 20.91012860925693687140590951899, 21.45012472530144185431573677918, 22.22014956618461647322420981624

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