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Properties

Label 2-54e2-108.43-c0-0-4
Degree $2$
Conductor $2916$
Sign $0.0581 - 0.998i$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)5-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.300i)10-s + (1.17 − 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.939 + 1.62i)17-s + (0.0603 − 0.342i)20-s + (−0.673 − 0.565i)25-s + 1.53·26-s + (1.17 + 0.984i)29-s + (−0.939 − 0.342i)32-s + (−0.326 + 1.85i)34-s + (0.939 + 1.62i)37-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.326 − 0.118i)5-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.300i)10-s + (1.17 − 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.939 + 1.62i)17-s + (0.0603 − 0.342i)20-s + (−0.673 − 0.565i)25-s + 1.53·26-s + (1.17 + 0.984i)29-s + (−0.939 − 0.342i)32-s + (−0.326 + 1.85i)34-s + (0.939 + 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.0581 - 0.998i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.0581 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845959730\)
\(L(\frac12)\) \(\approx\) \(1.845959730\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 \)
good5 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (-0.766 + 0.642i)T^{2} \)
13 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.680311212202106228079081221917, −8.102155496630937668751362197512, −7.87516178127700217126429971141, −6.49928494534750452299136319308, −6.21916130117342225134644867438, −5.30935972904565801870110177803, −4.50115684099798110655080065930, −3.55650321211864853639698654717, −3.07858315278118154206871361441, −1.49808106940792538028767976470, 1.02252612821314987703932933251, 2.22893309614917424960348012698, 3.21240028315476625980928729144, 3.95216250878845361058640131162, 4.71380791865590756978656657697, 5.61479704654847948146472390457, 6.32060866389577926107170634305, 7.12718470216379419976944090230, 7.923699415097293551346993797464, 9.044125838527562519982906907612

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