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Properties

Label 1-712-712.485-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.106 + 0.994i$
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  + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.540 + 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (−0.755 − 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.540 + 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (−0.755 − 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4311896728 + 0.4796839787i\)
\(L(\frac12)\) \(\approx\) \(0.4311896728 + 0.4796839787i\)
\(L(1)\) \(\approx\) \(0.8481692809 + 0.01200089599i\)
\(L(1)\) \(\approx\) \(0.8481692809 + 0.01200089599i\)

Euler product

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

−22.21745804958980675355865683994, −21.62674552477554608428516936641, −20.71559024516153543881031581193, −19.827767186249077438288638841236, −19.47407700007254524462537434760, −18.57106275663465377840943749891, −17.17707011208822565443575892041, −16.340290848414717362908352031736, −16.01149182267857771296676914005, −14.985174701112223148118658024901, −14.419801550968696961441736409840, −13.16555835789823796143722585664, −12.46932551931656101590263739092, −11.58335822279252586808361778390, −10.540873974264122684691714909039, −9.621792184571375625874599856838, −8.835230047344530914855062140190, −8.30343338676672960984676389963, −7.19505383555501909688521051872, −5.73875668625407864730271034419, −5.13221662331149652476013864301, −3.844193029520578307377335776, −3.360832272141555077479304537559, −2.13624164448060701465151341962, −0.27906572772228928278306253746, 1.473166450371630523237228850083, 2.5219168129984070716052905953, 3.6565824521848402283016096478, 4.17632887812353029414220508862, 6.07887246010403094271678229272, 6.75772711300825924594024024185, 7.45534620108882922297813999768, 8.14722081264530840192680169104, 9.49828921391035493246292550793, 10.034596370119988570646838100631, 11.36865555690209438379622244616, 12.09132938713806483410307799044, 12.83656637740376257356851399781, 13.81726695569923171914195872189, 14.652619662639897223330260913129, 15.03356935143424479187672569833, 16.427790662040412900970383893506, 17.07242975236330088235640434639, 18.1627643290520642699475456730, 18.935460184276839943781006288291, 19.549957295989093534470104590465, 19.992678549163127987049333647268, 21.084024558874128494799236466794, 22.22673979719077182256032717060, 22.97744782409231891388495662015

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