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Properties

Label 2-54e2-108.79-c0-0-6
Degree $2$
Conductor $2916$
Sign $0.973 + 0.230i$
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 + (0.766 − 0.642i)5-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + (−0.939 + 0.342i)20-s + 0.999·26-s + (0.173 − 0.984i)29-s + (−0.766 + 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)37-s + (−0.173 − 0.984i)40-s + (−0.347 − 1.96i)41-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + (−0.939 + 0.342i)20-s + 0.999·26-s + (0.173 − 0.984i)29-s + (−0.766 + 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)37-s + (−0.173 − 0.984i)40-s + (−0.347 − 1.96i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.973 + 0.230i$
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: \((2,\ 2916,\ (\ :0),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064887978\)
\(L(\frac12)\) \(\approx\) \(1.064887978\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.870829885855735537162450002072, −8.139649788344755399325537550834, −7.44772079885966836982933363170, −6.61114297046771494391776558120, −5.83422271247204906594377505536, −5.20234258147013821037343622256, −4.60044101755859539743321764834, −3.44645103851411782943988116695, −2.12878118604263574404422614148, −0.74179210762313269001488589162, 1.50449486110804725932240301981, 2.28886309346929892699716649621, 3.14749794923784185250832544853, 4.14929290520692569697587364461, 4.87179718169234604318532908511, 5.97051836024638153827705084235, 6.61696224540880936693594643216, 7.57264499305064097799184184068, 8.479037497920024770950131687864, 9.144653773711769825440086233323

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