Properties

Label 2-54e2-27.23-c0-0-2
Degree $2$
Conductor $2916$
Sign $-0.918 + 0.396i$
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)7-s + (−1.87 + 0.684i)13-s + (−1 − 1.73i)19-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.5 − 0.866i)37-s + (−0.173 − 0.984i)43-s + (−0.766 + 0.642i)61-s + (0.939 − 0.342i)67-s + (0.5 + 0.866i)73-s + (−1.87 − 0.684i)79-s + (1 − 1.73i)91-s + (0.347 + 1.96i)97-s + (−0.173 + 0.984i)103-s − 109-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)7-s + (−1.87 + 0.684i)13-s + (−1 − 1.73i)19-s + (−0.939 − 0.342i)25-s + (−0.766 − 0.642i)31-s + (0.5 − 0.866i)37-s + (−0.173 − 0.984i)43-s + (−0.766 + 0.642i)61-s + (0.939 − 0.342i)67-s + (0.5 + 0.866i)73-s + (−1.87 − 0.684i)79-s + (1 − 1.73i)91-s + (0.347 + 1.96i)97-s + (−0.173 + 0.984i)103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $-0.918 + 0.396i$
Analytic conductor: \(1.45527\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :0),\ -0.918 + 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1151078501\)
\(L(\frac12)\) \(\approx\) \(0.1151078501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.874631088111875141895908797186, −7.74522476189716614161888983229, −7.07017604011346862679643946489, −6.42750829390092320963302703394, −5.53324960843850486743908817212, −4.71851718919101419715562853271, −3.92762110810703715687938416074, −2.59508240641849098155244636314, −2.23554933301303663513992845668, −0.06471887733229646039410735002, 1.72469264365200716840682560887, 2.85538139248227052526624223090, 3.70202208893669596103443204989, 4.55590576807730731994737253149, 5.48017354533643088186369331710, 6.25797183390473037997142081541, 7.08563661800353171904027192304, 7.73232745596979637609867028638, 8.361156591932937764180847759542, 9.545712196485525307583088487005

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