Properties

Label 2-3312-3312.2437-c0-0-2
Degree $2$
Conductor $3312$
Sign $0.675 - 0.737i$
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  + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (0.642 + 0.766i)12-s + (1.58 + 0.424i)13-s + (0.173 + 0.984i)16-s + (−0.642 + 0.766i)18-s + (−0.866 + 0.5i)23-s + (−0.939 + 0.342i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 1.34i)26-s + (−0.866 − 0.5i)27-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.5 − 0.866i)6-s + (0.866 − 0.500i)8-s + (0.939 + 0.342i)9-s + (0.642 + 0.766i)12-s + (1.58 + 0.424i)13-s + (0.173 + 0.984i)16-s + (−0.642 + 0.766i)18-s + (−0.866 + 0.5i)23-s + (−0.939 + 0.342i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 1.34i)26-s + (−0.866 − 0.5i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $0.675 - 0.737i$
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: \((2,\ 3312,\ (\ :0),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7341006346\)
\(L(\frac12)\) \(\approx\) \(0.7341006346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.984 + 0.173i)T \)
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 \( 1 + (-1.58 - 0.424i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (0.515 + 1.92i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.642 - 1.11i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.28iT - T^{2} \)
73 \( 1 - 1.53iT - T^{2} \)
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

−8.747166653123590568092426059213, −8.059873326480219841095755730893, −7.34094073367821942922870710302, −6.58662513075588384286457942994, −5.80897545005345762631150389525, −5.66145649017699047505802708828, −4.24724949300625033954846303522, −3.99536681908487270651279924024, −2.00151341934609961157236571258, −0.861637822864739925262310365618, 0.885702225483991079135124852995, 1.87719430107657306337371474436, 3.24265916019451836356391656791, 3.97778488682397755181620402242, 4.69575024660133219856958211174, 5.74310074651202867055692940346, 6.21779829994739139412514945250, 7.40872901200970374651820410112, 8.034573740776755519197876621127, 8.951595134149846155823964962994

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