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Properties

Label 1-712-712.5-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.555 - 0.831i$
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  + (−0.281 + 0.959i)3-s + (0.415 + 0.909i)5-s + (−0.909 + 0.415i)7-s + (−0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.281 − 0.959i)13-s + (−0.989 + 0.142i)15-s + (0.142 − 0.989i)17-s + (−0.540 + 0.841i)19-s + (−0.142 − 0.989i)21-s + (−0.540 + 0.841i)23-s + (−0.654 + 0.755i)25-s + (0.755 − 0.654i)27-s + (−0.909 + 0.415i)29-s + (−0.540 − 0.841i)31-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)3-s + (0.415 + 0.909i)5-s + (−0.909 + 0.415i)7-s + (−0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.281 − 0.959i)13-s + (−0.989 + 0.142i)15-s + (0.142 − 0.989i)17-s + (−0.540 + 0.841i)19-s + (−0.142 − 0.989i)21-s + (−0.540 + 0.841i)23-s + (−0.654 + 0.755i)25-s + (0.755 − 0.654i)27-s + (−0.909 + 0.415i)29-s + (−0.540 − 0.841i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1461890771 + 0.2734318221i\)
\(L(\frac12)\) \(\approx\) \(-0.1461890771 + 0.2734318221i\)
\(L(1)\) \(\approx\) \(0.5913351653 + 0.3829353628i\)
\(L(1)\) \(\approx\) \(0.5913351653 + 0.3829353628i\)

Euler product

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

−21.95391658589791548626353727230, −21.4095561664601078462515525929, −20.23708743929627071620669786467, −19.55874072744277685436478542013, −18.87722596222796523943209032571, −18.04585796433458336938935762111, −16.88358963579257299971769892380, −16.70537759161675832716828663041, −15.78950419075264191481726310680, −14.30701487397596768458503638449, −13.55594262568465173654659265819, −12.94986271362709655537525500876, −12.38360321750401885771231694196, −11.251283689161453532188519374770, −10.44791990218148795498390985572, −9.192624226041596603055066437372, −8.56080742567143864392497613196, −7.58426602361037614717664287134, −6.36228922653044967807393560294, −6.06755108676641170098447721722, −4.84557765698301519296174752161, −3.68147745222083930371739787660, −2.380935969673286404929942655748, −1.36692337681245281787008240144, −0.1487552814623217238071425555, 2.12677467810815140398424580257, 3.13606338503143907214714832583, 3.80518318097986999740054297281, 5.26283900879891489061952690040, 5.81419246131036894159080265522, 6.79903083927598789286510770408, 7.83621113322698264628357826546, 9.174259252740273862229331649868, 9.9082620224559477589790360247, 10.33524304331530218038597627146, 11.32677595420090949425513225204, 12.27800932450580437830500444437, 13.23243447385897060114258437599, 14.230854238612571536210954324492, 15.294626648548028951350198814976, 15.44195706739479096867139314279, 16.55294481653045203682698274294, 17.40031736290279283746856649148, 18.27354595417318358702943557799, 18.86874145532042595571808087199, 20.21670514684324454864153250544, 20.62171025276427512974995409560, 21.75771485457339133986334703854, 22.329155082958344564040038167053, 22.85118538837622539412412134526

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