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Properties

Label 2-3312-3312.2437-c0-0-4
Degree $2$
Conductor $3312$
Sign $0.300 + 0.953i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
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  + (0.984 − 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (0.168 + 0.0451i)13-s + (0.766 − 0.642i)16-s + (−0.342 − 0.939i)18-s + (−0.866 + 0.5i)23-s + (0.173 − 0.984i)24-s + (−0.866 − 0.5i)25-s + (0.173 + 0.0151i)26-s + (−0.866 − 0.500i)27-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.5i)8-s + (−0.173 − 0.984i)9-s + (0.342 − 0.939i)12-s + (0.168 + 0.0451i)13-s + (0.766 − 0.642i)16-s + (−0.342 − 0.939i)18-s + (−0.866 + 0.5i)23-s + (0.173 − 0.984i)24-s + (−0.866 − 0.5i)25-s + (0.173 + 0.0151i)26-s + (−0.866 − 0.500i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $0.300 + 0.953i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (2437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 0.300 + 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.935561773\)
\(L(\frac12)\) \(\approx\) \(2.935561773\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.168 - 0.0451i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-0.218 - 0.816i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.342 - 0.592i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
73 \( 1 + 1.87iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.479653028702486284196404879849, −7.75445347153901619673788237153, −7.12883296187452947691125793638, −6.31267846631638790543048067649, −5.79240677077254261508547793797, −4.71935788466202249543917713119, −3.83302347412319479790763661983, −3.10953711588330297769498492023, −2.20504659469396189218985496080, −1.32515331306439125526432322375, 1.92572654510874469585405876638, 2.69530212330141873568519261911, 3.71028137350125432698306437660, 4.16412286343482086496124653282, 5.06150137196747858383844708161, 5.77636475693141378260426387986, 6.56854693480337455583224656711, 7.57736478727165845555166798402, 8.096312443324484191823534211672, 8.834009063408822384551823442787

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