Properties

Label 1-756-756.31-r0-0-0
Degree $1$
Conductor $756$
Sign $0.986 - 0.163i$
Analytic cond. $3.51084$
Root an. cond. $3.51084$
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  + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + 37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s + (−0.766 − 0.642i)11-s + (−0.766 + 0.642i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + 37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + (−0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(3.51084\)
Root analytic conductor: \(3.51084\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 756,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050040736 - 0.08636968743i\)
\(L(\frac12)\) \(\approx\) \(1.050040736 - 0.08636968743i\)
\(L(1)\) \(\approx\) \(0.8949462433 + 0.02033281264i\)
\(L(1)\) \(\approx\) \(0.8949462433 + 0.02033281264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−22.69031764132102450585030674907, −21.4576424383489268591508244907, −20.81465640293848713207726850097, −20.037554262973374132544154272161, −19.279055313877676004296726832953, −18.565840636074190638472232857190, −17.35406053325366942708342554531, −16.913944969830252294927756170267, −15.80638309092145009238182358130, −15.22038167141519817877380060170, −14.45731501750874936522371932302, −13.1554900931028617641407370997, −12.49588939320423589606779501828, −11.99325168333595774847212388627, −10.67796690290755150784617797081, −10.10722788603953010898194393556, −8.94307005848624988088321833168, −7.99463228802874565887666962093, −7.55508642579575674164407834789, −6.2630830231297309527575720400, −5.139505908197421446174401205044, −4.48093371361693892407696186330, −3.38584392941519654647973628110, −2.26157150188645011251361432063, −0.87332723381316308490684913930, 0.69206042953961884899030378921, 2.525003344853591956818323243803, 3.0748503968885031031217366451, 4.35454964187152723403422974336, 5.14876644492015972734372629388, 6.43225439585168521004576363404, 7.23318361292040852166103969556, 7.96441903759052196412416030151, 8.98723777684048492384973832021, 9.97400354507083781251256905154, 10.96120351503651622421404523190, 11.51975996803195800581414130509, 12.4285366751230884957048158700, 13.459993505596638017441537640, 14.28171097241493945640727933777, 15.11205301617159300462619066740, 15.8447233166532382713920375511, 16.621185189813430252473320853897, 17.6007948223664638835813056802, 18.60136792389434805115011501568, 19.09015794656428557084275272302, 19.82864102566240426652984707451, 20.86811606848765526369185694549, 21.66807821883583138162046672913, 22.364891663827790373807932604079

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