Properties

Label 2-54e2-27.2-c0-0-0
Degree $2$
Conductor $2916$
Sign $0.802 - 0.597i$
Analytic cond. $1.45527$
Root an. cond. $1.20634$
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)7-s + (0.347 + 1.96i)13-s + (−1 + 1.73i)19-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (0.5 + 0.866i)37-s + (−0.766 − 0.642i)43-s + (0.939 − 0.342i)61-s + (−0.173 − 0.984i)67-s + (0.5 − 0.866i)73-s + (0.347 − 1.96i)79-s + (1 + 1.73i)91-s + (1.53 + 1.28i)97-s + (−0.766 + 0.642i)103-s − 109-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)7-s + (0.347 + 1.96i)13-s + (−1 + 1.73i)19-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (0.5 + 0.866i)37-s + (−0.766 − 0.642i)43-s + (0.939 − 0.342i)61-s + (−0.173 − 0.984i)67-s + (0.5 − 0.866i)73-s + (0.347 − 1.96i)79-s + (1 + 1.73i)91-s + (1.53 + 1.28i)97-s + (−0.766 + 0.642i)103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.802 - 0.597i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.802 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353484053\)
\(L(\frac12)\) \(\approx\) \(1.353484053\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.851413444658906576718447240319, −8.288419252843290799278437800192, −7.66002456504570874270782741849, −6.52900816339452302005399898577, −6.27298097548638133773622895935, −4.93065436160485090905542490572, −4.36677181808462205386167842181, −3.63763504124611025364846249104, −2.16338642521178496328017446251, −1.47517846044465817875606327579, 0.936576595728922874905163411600, 2.32072647263946060697173543796, 3.08797613226390450602248325129, 4.24677184440665543676023549659, 5.10736910415855096389836453000, 5.62968299356973887661820237208, 6.61661737475512348013932356994, 7.46630848729481122130193368983, 8.267120794677861144762326417496, 8.611413995363210587204824987059

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