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Properties

Label 2-54e2-108.79-c0-0-3
Degree $2$
Conductor $2916$
Sign $0.893 - 0.448i$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−1.17 + 0.984i)5-s + (0.5 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.326 − 1.85i)13-s + (0.766 + 0.642i)16-s + (0.173 + 0.300i)17-s + (1.43 − 0.524i)20-s + (0.233 − 1.32i)25-s + 1.87·26-s + (0.326 − 1.85i)29-s + (−0.766 + 0.642i)32-s + (−0.326 + 0.118i)34-s + (−0.173 − 0.300i)37-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−1.17 + 0.984i)5-s + (0.5 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.326 − 1.85i)13-s + (0.766 + 0.642i)16-s + (0.173 + 0.300i)17-s + (1.43 − 0.524i)20-s + (0.233 − 1.32i)25-s + 1.87·26-s + (0.326 − 1.85i)29-s + (−0.766 + 0.642i)32-s + (−0.326 + 0.118i)34-s + (−0.173 − 0.300i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (2107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6758025605\)
\(L(\frac12)\) \(\approx\) \(0.6758025605\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.567734062570606727845127220398, −8.069010507219445904201872790362, −7.54573521003064845565362582492, −6.91982401257187958500282798538, −6.03087550011020012834958717038, −5.35142130331332503304518895476, −4.27928498706815031881169771930, −3.58527248132003980107588319217, −2.66202236774022975209342817265, −0.57845342897278783417822361580, 1.03910765909590387245671992510, 2.11814454393874900913500871597, 3.36556095980728534996358513472, 4.15717372399233684942562184467, 4.66341246615592086004609433665, 5.44385801849725248041623338955, 6.92720756069266168254475863383, 7.49097836434615686397530093190, 8.525407177059857270903483392543, 8.855481140779454065310045389048

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