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Properties

Label 1-712-712.43-r0-0-0
Degree $1$
Conductor $712$
Sign $0.0173 + 0.999i$
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  + (−0.479 + 0.877i)3-s + (−0.909 − 0.415i)5-s + (0.349 + 0.936i)7-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (0.877 + 0.479i)13-s + (0.800 − 0.599i)15-s + (0.989 − 0.142i)17-s + (−0.977 − 0.212i)19-s + (−0.989 − 0.142i)21-s + (−0.212 + 0.977i)23-s + (0.654 + 0.755i)25-s + (0.997 − 0.0713i)27-s + (0.936 − 0.349i)29-s + (−0.212 − 0.977i)31-s + ⋯
L(s)  = 1  + (−0.479 + 0.877i)3-s + (−0.909 − 0.415i)5-s + (0.349 + 0.936i)7-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (0.877 + 0.479i)13-s + (0.800 − 0.599i)15-s + (0.989 − 0.142i)17-s + (−0.977 − 0.212i)19-s + (−0.989 − 0.142i)21-s + (−0.212 + 0.977i)23-s + (0.654 + 0.755i)25-s + (0.997 − 0.0713i)27-s + (0.936 − 0.349i)29-s + (−0.212 − 0.977i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0173 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0173 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0173 + 0.999i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.0173 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6291280777 + 0.6401141043i\)
\(L(\frac12)\) \(\approx\) \(0.6291280777 + 0.6401141043i\)
\(L(1)\) \(\approx\) \(0.7587615072 + 0.2677315116i\)
\(L(1)\) \(\approx\) \(0.7587615072 + 0.2677315116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.479 + 0.877i)T \)
5 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (0.349 + 0.936i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (0.877 + 0.479i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (-0.977 - 0.212i)T \)
23 \( 1 + (-0.212 + 0.977i)T \)
29 \( 1 + (0.936 - 0.349i)T \)
31 \( 1 + (-0.212 - 0.977i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.877 - 0.479i)T \)
43 \( 1 + (-0.936 - 0.349i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (0.479 + 0.877i)T \)
61 \( 1 + (0.0713 + 0.997i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (0.800 + 0.599i)T \)
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

−23.037194369587797739677313910084, −21.64015882331098718967304522307, −20.548868889837577258099721331531, −19.8884666453925592259026440785, −19.116771139439289925818331388546, −18.20021547789443131557341085907, −17.76193298579624354126568454960, −16.64439979890020368648781983146, −16.05007121967070427143388089370, −14.78100137622638114108558538532, −14.25638015540146080218141386699, −13.03195166651747655077080814754, −12.5162968580745859650835393433, −11.57721457262208793972449863437, −10.68041840457520848917780575784, −10.268524109185012224302783061253, −8.34508104610192246931011056876, −7.95332660241076178782920161963, −7.031829073034464992263399728302, −6.386040350023462273412854882642, −5.079787475873869251324577918155, −4.163337841455238239077860646783, −3.056748847269945179078705229060, −1.75004790067064434602484535589, −0.58002410803321802459507212136, 1.04330049126396581400689894279, 2.79298622627522652540974149754, 3.76892599552267658901519370537, 4.58967861606987782753023650111, 5.5886748924176484262242322572, 6.19824993551569845466713602002, 7.77852339010321577353987671405, 8.56972531051335400796679735759, 9.18501695016446217165928776744, 10.36870145260117831016141598993, 11.38810515524437213263864994107, 11.6503218489142497825757550105, 12.644058375493058083167769504377, 13.79675404096584855365546042144, 14.947736661388906337085163901410, 15.480944888301841907522003017100, 16.24181118631022550143257425209, 16.778722570877837019003484091724, 17.927330463087022259618031516809, 18.804702560678852063627276635627, 19.46395863043777169779901759316, 20.72045961326133966689203276110, 21.18773905793055446124975740733, 21.81518234742795732817463309526, 22.856343819371098027731938532078

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