Properties

Label 2-3312-3312.2437-c0-0-5
Degree $2$
Conductor $3312$
Sign $-0.843 - 0.537i$
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  + (0.642 − 0.766i)2-s + (0.173 − 0.984i)3-s + (−0.173 − 0.984i)4-s + (−0.642 − 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s − 12-s + (−1.10 − 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (−0.866 + 0.5i)23-s + (−0.642 + 0.766i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 0.657i)26-s + (−0.5 + 0.866i)27-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (0.173 − 0.984i)3-s + (−0.173 − 0.984i)4-s + (−0.642 − 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s − 12-s + (−1.10 − 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (−0.866 + 0.5i)23-s + (−0.642 + 0.766i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 0.657i)26-s + (−0.5 + 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $-0.843 - 0.537i$
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: \((2,\ 3312,\ (\ :0),\ -0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201044540\)
\(L(\frac12)\) \(\approx\) \(1.201044540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
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 \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (0.0451 + 0.168i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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

−8.289323430700708090402444899517, −7.56537794847869749166356936423, −6.87343198004343168787301148949, −5.83745694012777582669235384648, −5.53809646955414984395217723419, −4.33038565878221517071587559923, −3.55145603458386748570029644950, −2.42148530167617330499323008341, −2.00931072359312939254148980709, −0.51779728772103489763958206303, 2.30559670707499222327654737909, 3.11484178502334324256748151399, 4.12614550354495853281006534681, 4.54447724802833425419431304318, 5.48649353698896184449417528878, 5.98448577331832638682053041467, 7.07108441249072634804424649391, 7.68645379013592894576874928498, 8.462655695451532459853425323193, 9.207437767888112689484546169721

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