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Properties

Label 1-768-768.5-r1-0-0
Degree $1$
Conductor $768$
Sign $-0.449 - 0.893i$
Analytic cond. $82.5330$
Root an. cond. $82.5330$
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  + (−0.995 − 0.0980i)5-s + (0.555 + 0.831i)7-s + (0.471 − 0.881i)11-s + (−0.0980 − 0.995i)13-s + (0.923 − 0.382i)17-s + (−0.634 + 0.773i)19-s + (−0.195 − 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.881 + 0.471i)29-s + (0.707 + 0.707i)31-s + (−0.471 − 0.881i)35-s + (−0.773 + 0.634i)37-s + (−0.980 + 0.195i)41-s + (0.956 − 0.290i)43-s + (0.382 + 0.923i)47-s + ⋯
L(s)  = 1  + (−0.995 − 0.0980i)5-s + (0.555 + 0.831i)7-s + (0.471 − 0.881i)11-s + (−0.0980 − 0.995i)13-s + (0.923 − 0.382i)17-s + (−0.634 + 0.773i)19-s + (−0.195 − 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.881 + 0.471i)29-s + (0.707 + 0.707i)31-s + (−0.471 − 0.881i)35-s + (−0.773 + 0.634i)37-s + (−0.980 + 0.195i)41-s + (0.956 − 0.290i)43-s + (0.382 + 0.923i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.449 - 0.893i$
Analytic conductor: \(82.5330\)
Root analytic conductor: \(82.5330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 768,\ (1:\ ),\ -0.449 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5132179638 - 0.8329000610i\)
\(L(\frac12)\) \(\approx\) \(0.5132179638 - 0.8329000610i\)
\(L(1)\) \(\approx\) \(0.8901434975 - 0.09456266488i\)
\(L(1)\) \(\approx\) \(0.8901434975 - 0.09456266488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.995 - 0.0980i)T \)
7 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (0.471 - 0.881i)T \)
13 \( 1 + (-0.0980 - 0.995i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (-0.634 + 0.773i)T \)
23 \( 1 + (-0.195 - 0.980i)T \)
29 \( 1 + (-0.881 + 0.471i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.773 + 0.634i)T \)
41 \( 1 + (-0.980 + 0.195i)T \)
43 \( 1 + (0.956 - 0.290i)T \)
47 \( 1 + (0.382 + 0.923i)T \)
53 \( 1 + (-0.881 - 0.471i)T \)
59 \( 1 + (0.0980 - 0.995i)T \)
61 \( 1 + (-0.290 + 0.956i)T \)
67 \( 1 + (0.290 - 0.956i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.555 - 0.831i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (0.773 + 0.634i)T \)
89 \( 1 + (-0.195 + 0.980i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−22.63105181193502296253771117029, −21.56508250389306235187472389805, −20.76146516110748157132763009572, −19.9653342026784151073548783803, −19.3062963015411517027041396825, −18.59046598675432241168758609727, −17.24831048808240809853466495486, −17.05713513823224776467653152050, −15.87130074935768241491224275140, −15.06494751554943091986885323214, −14.40124035672916739251270267881, −13.52060241096740062241384890136, −12.40618844095100394918606916179, −11.67524566509588212081009173971, −11.016128098698784189418315585311, −10.01312855010589943754088993773, −9.07296611188241284812049259455, −7.97515052087791186136650624558, −7.33079825373434509608758310501, −6.61464572972840394907575350915, −5.15845016574107448587595044118, −4.15894562739053836631928217332, −3.76651864842247856085125711671, −2.17776621592906999951228360351, −1.09039890896863012651761459745, 0.251679086181678493514857204889, 1.42386366056958733323562086657, 2.88030850529429984802173033577, 3.62175786933868080477663294556, 4.805567584322457088836954704760, 5.61866131837314129626332246543, 6.63230380736482629812204604667, 7.972720540175835220971453734068, 8.2407779203324975681855423556, 9.21083856278070351634122936135, 10.49781800266043625735867229144, 11.17939015281657934810778449276, 12.26634701625170516500201711028, 12.41092201642984536933616891168, 13.90952366782115387550359410951, 14.7015123639837397692107192228, 15.34578565975105347454795929584, 16.23165070409274620518915397111, 16.92163232021387932698264545708, 18.0292037545632659617591764366, 18.88306025790593978237276627321, 19.26236572879543813567726949799, 20.494569427672225628761870245389, 20.91126184681673337761784755319, 22.10247926912040295370368813364

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