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Properties

Label 2-3312-3312.229-c0-0-1
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 + (−0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (−0.866 − 0.500i)8-s + (−0.766 − 0.642i)9-s + (0.984 + 0.173i)12-s + (0.218 + 0.816i)13-s + (−0.939 + 0.342i)16-s + (−0.984 + 0.173i)18-s + (0.866 + 0.5i)23-s + (0.766 − 0.642i)24-s + (0.866 − 0.5i)25-s + (0.766 + 0.357i)26-s + (0.866 − 0.500i)27-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (−0.866 − 0.500i)8-s + (−0.766 − 0.642i)9-s + (0.984 + 0.173i)12-s + (0.218 + 0.816i)13-s + (−0.939 + 0.342i)16-s + (−0.984 + 0.173i)18-s + (0.866 + 0.5i)23-s + (0.766 − 0.642i)24-s + (0.866 − 0.5i)25-s + (0.766 + 0.357i)26-s + (0.866 − 0.500i)27-s + ⋯

Functional equation

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

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: $\chi_{3312} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.521048006\)
\(L(\frac12)\) \(\approx\) \(1.521048006\)
\(L(1)\) 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 \( 1 + (0.342 - 0.939i)T \)
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 \( 1 + (-0.218 - 0.816i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-1.58 - 0.424i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.984 + 1.70i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - 1.96iT - T^{2} \)
73 \( 1 - 0.347iT - T^{2} \)
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

−8.996048192566108651461975871722, −8.326738083359740284495473458204, −6.91063208292355619564616347125, −6.27820495809226379899819957526, −5.51288386578881329021390295271, −4.62181145848624574113184272603, −4.26466357715707346144177685828, −3.23354058627188992796032497560, −2.49910909949959227996014913641, −1.02015935080939403252137208539, 1.08128083431905731818197378847, 2.69406346189827164790691128921, 3.18265629313491393474914054428, 4.65872669921425809335864023954, 5.06485586869099412253191630402, 6.13765503682641039852427523310, 6.46195722952352150757454338592, 7.32055144061864209632778579661, 7.919716896605766222073492589073, 8.576263047464021851793316181415

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