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Properties

Label 1-712-712.373-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.837 + 0.547i$
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  + (−0.909 − 0.415i)3-s + (−0.142 + 0.989i)5-s + (−0.989 − 0.142i)7-s + (0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.540 − 0.841i)15-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (0.841 + 0.540i)21-s + (0.755 − 0.654i)23-s + (−0.959 − 0.281i)25-s + (−0.281 − 0.959i)27-s + (−0.989 − 0.142i)29-s + (0.755 + 0.654i)31-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)3-s + (−0.142 + 0.989i)5-s + (−0.989 − 0.142i)7-s + (0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.540 − 0.841i)15-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (0.841 + 0.540i)21-s + (0.755 − 0.654i)23-s + (−0.959 − 0.281i)25-s + (−0.281 − 0.959i)27-s + (−0.989 − 0.142i)29-s + (0.755 + 0.654i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.837 + 0.547i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.837 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1321993815 + 0.4439597249i\)
\(L(\frac12)\) \(\approx\) \(0.1321993815 + 0.4439597249i\)
\(L(1)\) \(\approx\) \(0.6199557966 + 0.1563919521i\)
\(L(1)\) \(\approx\) \(0.6199557966 + 0.1563919521i\)

Euler product

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

−22.40491374323307743726487663788, −21.483535096217042067324086785456, −20.73160700089740492434122721133, −19.96934886914077706329737027765, −18.91600999219715990428938174123, −18.193519721507609620354930813839, −17.08446299192226633651116201, −16.529008176270750674133074008509, −15.83497117050348337632860964915, −15.36485226810219219407815370590, −13.65801132164830060209341455768, −13.14210840294337005245110423291, −12.21997584924795359926396104192, −11.44209499208324154333815371008, −10.6741835677246564157649527991, −9.46168235242802224382841647350, −9.07857646783633508797252156551, −7.86856849204030185346831607188, −6.600377198971914958451423976085, −5.803428249750622770510025484229, −5.16689086471081033915532917235, −3.9438497407785746939482262171, −3.250924189261550806614933650498, −1.35199023036422018236486784395, −0.27436912101501187830714809109, 1.45196168046861910336295726116, 2.66069758680338994999753223447, 3.78277439212362409777370259790, 4.814471536729411783689583908, 6.07909564321981634919718804932, 6.8293790723377158320281428560, 7.08982241418842051532273255862, 8.53094502018286500816008354633, 9.80194331230411057543384034248, 10.42804110390146461321589460413, 11.3382858449165788718901022114, 11.9652223989943131735128102287, 13.1680033831063429293573260720, 13.475896802427436953013195090596, 14.9010684868099577125554078462, 15.58532570305262668847151208572, 16.43379964754689755052524066163, 17.31897381270596847529955485639, 18.12184069170574978448286069988, 18.728050680750127262683867285493, 19.49462390063511664135572333556, 20.37042968118915201987662942710, 21.6962737691369922002661398663, 22.28359932496457818991610085757, 22.97505677175327085739114172961

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