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Properties

Label 1-712-712.13-r1-0-0
Degree $1$
Conductor $712$
Sign $0.584 + 0.811i$
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  + (0.0713 − 0.997i)3-s + (0.281 − 0.959i)5-s + (0.479 + 0.877i)7-s + (−0.989 − 0.142i)9-s + (−0.959 + 0.281i)11-s + (−0.997 − 0.0713i)13-s + (−0.936 − 0.349i)15-s + (−0.909 − 0.415i)17-s + (−0.599 − 0.800i)19-s + (0.909 − 0.415i)21-s + (0.800 − 0.599i)23-s + (−0.841 − 0.540i)25-s + (−0.212 + 0.977i)27-s + (0.877 − 0.479i)29-s + (0.800 + 0.599i)31-s + ⋯
L(s)  = 1  + (0.0713 − 0.997i)3-s + (0.281 − 0.959i)5-s + (0.479 + 0.877i)7-s + (−0.989 − 0.142i)9-s + (−0.959 + 0.281i)11-s + (−0.997 − 0.0713i)13-s + (−0.936 − 0.349i)15-s + (−0.909 − 0.415i)17-s + (−0.599 − 0.800i)19-s + (0.909 − 0.415i)21-s + (0.800 − 0.599i)23-s + (−0.841 − 0.540i)25-s + (−0.212 + 0.977i)27-s + (0.877 − 0.479i)29-s + (0.800 + 0.599i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5764123022 + 0.2950417286i\)
\(L(\frac12)\) \(\approx\) \(0.5764123022 + 0.2950417286i\)
\(L(1)\) \(\approx\) \(0.8343634969 - 0.3151713194i\)
\(L(1)\) \(\approx\) \(0.8343634969 - 0.3151713194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.0713 - 0.997i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.479 + 0.877i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.997 - 0.0713i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (-0.599 - 0.800i)T \)
23 \( 1 + (0.800 - 0.599i)T \)
29 \( 1 + (0.877 - 0.479i)T \)
31 \( 1 + (0.800 + 0.599i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.997 + 0.0713i)T \)
43 \( 1 + (0.877 + 0.479i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (-0.0713 - 0.997i)T \)
61 \( 1 + (-0.977 - 0.212i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (0.281 + 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.989 - 0.142i)T \)
83 \( 1 + (-0.936 + 0.349i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−22.1905215756745999419959154151, −21.36327017863507489420953356053, −20.94277305471448506209227972413, −19.854738984663975656346569619001, −19.18557837976299028075461495139, −18.04463607506995132897305839318, −17.26281709319872449225088576213, −16.692738131683148151631116271231, −15.3971800495820352698918597002, −15.07100099554405576795070531900, −14.02959178059279885196211897882, −13.5631163868578042817376013855, −12.147605189428535458130024638117, −10.96952486830729780882430403845, −10.57455925854034308524494368100, −9.97898217953223471579694497208, −8.82297467671694788443921700840, −7.81791655550426079401957655719, −6.961740836089914448346248876190, −5.81423783345329968531670949228, −4.86348846829179931391779543582, −3.983985859051454712043518593736, −2.98270780985739860305283880581, −2.06211250262560305618285932789, −0.15924030050503849601259788643, 0.95772353390762953481411838118, 2.293957765158177627567916454180, 2.610646082058074100246906561535, 4.78081737714944097465119554509, 5.07970287429966520695958802447, 6.28673534549274384839087041243, 7.198727785226691611908231756527, 8.32547851877004327162819039813, 8.68527931697315638260643182656, 9.751244786526407433973407586848, 11.01557132412589400940953255323, 12.00335121268319320547079243268, 12.57177313567651069419208555039, 13.248710644297764981415991591471, 14.096959673862014143226214089747, 15.18214286780609706442215895517, 15.84633897443242479271729598096, 17.290673375183915242997950673859, 17.46036992010228139867332361735, 18.43055510445853299449245029890, 19.240282979904655199948128978447, 20.049600114192273110878049000895, 20.84657562815378855424260955491, 21.5923669259533352494835498783, 22.588823632062467220194073550989

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