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Properties

Label 1-756-756.139-r0-0-0
Degree $1$
Conductor $756$
Sign $0.727 + 0.686i$
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  + (0.939 + 0.342i)5-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.5 − 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + 53-s + 55-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)25-s + (0.766 + 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.5 − 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(3.51084\)
Root analytic conductor: \(3.51084\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 756,\ (0:\ ),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.619477141 + 0.6433771009i\)
\(L(\frac12)\) \(\approx\) \(1.619477141 + 0.6433771009i\)
\(L(1)\) \(\approx\) \(1.267896144 + 0.2040858391i\)
\(L(1)\) \(\approx\) \(1.267896144 + 0.2040858391i\)

Euler product

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

−22.27434057085565090630455067210, −21.56842053196361611283179561874, −20.72882673420257835583567154992, −19.970229329078577636309376711512, −19.204360843720037775888316830702, −18.125599807639053130791897799471, −17.23804358327377632666539088109, −17.06305579297051213015382392552, −15.76753228123894878988963321632, −14.97088003619600973480796562437, −14.00667853183014198238680968633, −13.43308222416985433745651174236, −12.39797055755438821828373816606, −11.748018712368585969072656448479, −10.561599181552709785497403410871, −9.62929280056252669200015787890, −9.23422944205499243993433955109, −8.01815595937429149256136528333, −7.02736872819350687800908657727, −6.12622992320931341608372903168, −5.1918763994633095663754028984, −4.3929651918956760655312149051, −3.01030825414021102530353662765, −2.0777247360701094768941256166, −0.89422302596482003767852217901, 1.355678569075290145299737737973, 2.22760446823431233059408484063, 3.39999934180181695077366891605, 4.43587544513420636267653191237, 5.58675612415462286072487361758, 6.40214548976735752856590671324, 7.07895150201010772478634417367, 8.43903665353678517360815347257, 9.12440305726340128261592456120, 10.19107239987719473642671019602, 10.63270377201232878707186763572, 11.98123729353846688447416299891, 12.51313137839968420405017505292, 13.69791925479125974459646367122, 14.43981314803860041243911415426, 14.79383784721329197056179255089, 16.292786580692021037795867986302, 16.88994797947908746615327277499, 17.5631753030998246812710190433, 18.54123257404474339995676346558, 19.2410021285956890715881273931, 20.03668082423972150977505533002, 21.2471493648072981563992665811, 21.58664426873456559716562973915, 22.400167611469072378257769102119

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