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Properties

Label 2-54e2-108.79-c0-0-0
Degree $2$
Conductor $2916$
Sign $-0.835 - 0.549i$
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 + (−0.266 + 0.223i)5-s + (0.5 − 0.866i)8-s + (−0.173 − 0.300i)10-s + (0.266 + 1.50i)13-s + (0.766 + 0.642i)16-s + (−0.939 − 1.62i)17-s + (0.326 − 0.118i)20-s + (−0.152 + 0.866i)25-s − 1.53·26-s + (−0.266 + 1.50i)29-s + (−0.766 + 0.642i)32-s + (1.76 − 0.642i)34-s + (0.939 + 1.62i)37-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.266 + 0.223i)5-s + (0.5 − 0.866i)8-s + (−0.173 − 0.300i)10-s + (0.266 + 1.50i)13-s + (0.766 + 0.642i)16-s + (−0.939 − 1.62i)17-s + (0.326 − 0.118i)20-s + (−0.152 + 0.866i)25-s − 1.53·26-s + (−0.266 + 1.50i)29-s + (−0.766 + 0.642i)32-s + (1.76 − 0.642i)34-s + (0.939 + 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $-0.835 - 0.549i$
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: \((2,\ 2916,\ (\ :0),\ -0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7711219694\)
\(L(\frac12)\) \(\approx\) \(0.7711219694\)
\(L(1)\) 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 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.198842844695328907268255963944, −8.516778283667631336587196997788, −7.57004365085250369138399747888, −6.90368260027238318873369123510, −6.55354809592184060052291701937, −5.42810596494073410879044402272, −4.69102855300883548667065622322, −4.00719159135747827772837884901, −2.85241929592816566108035659231, −1.39057912741315667406817591955, 0.55029062808196669594829843539, 1.95028615614210520978432154118, 2.81481039879339028291885898435, 3.96485024863266854847751969429, 4.29112006860368432031351500332, 5.57091736165754983755651844968, 6.08732091494311965658087890368, 7.48652958056246916341337686078, 8.109258398604285869477060037920, 8.639123722703947457986475051135

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