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Properties

Label 2-54e2-108.103-c0-0-5
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 + (1.76 − 0.642i)5-s + (−0.500 − 0.866i)8-s + (0.939 − 1.62i)10-s + (0.266 + 0.223i)13-s + (−0.939 − 0.342i)16-s + (−0.766 + 1.32i)17-s + (−0.326 − 1.85i)20-s + (1.93 − 1.62i)25-s + 0.347·26-s + (0.266 − 0.223i)29-s + (−0.939 + 0.342i)32-s + (0.266 + 1.50i)34-s + (−0.766 + 1.32i)37-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (1.76 − 0.642i)5-s + (−0.500 − 0.866i)8-s + (0.939 − 1.62i)10-s + (0.266 + 0.223i)13-s + (−0.939 − 0.342i)16-s + (−0.766 + 1.32i)17-s + (−0.326 − 1.85i)20-s + (1.93 − 1.62i)25-s + 0.347·26-s + (0.266 − 0.223i)29-s + (−0.939 + 0.342i)32-s + (0.266 + 1.50i)34-s + (−0.766 + 1.32i)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: $\chi_{2916} (1783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.0581 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.473672795\)
\(L(\frac12)\) \(\approx\) \(2.473672795\)
\(L(1)\) 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 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.927310255094790058808103723533, −8.333941333981557709931577871141, −6.71913444490560647262334487876, −6.33172766666468152463941805282, −5.58405422689924948617014296032, −4.91944278281353376886660435671, −4.12468987163546088852422192198, −2.96890861518123155901456172869, −1.95766241135129112534890215597, −1.42024497780840461471564395965, 1.87975804202794685931256537072, 2.68950724855969580616238508850, 3.45757593931483354762380606686, 4.81156196846572068427364157875, 5.30299630369616603304600768306, 6.14542304980556409782240417532, 6.67228864794800481425799553604, 7.26522963639163213077485214615, 8.308578376911450873817852807689, 9.209348056615209801576082449098

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