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Properties

Label 2-3312-3312.1885-c0-0-3
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  + (−0.984 + 0.173i)2-s + (−0.939 − 0.342i)3-s + (0.939 − 0.342i)4-s + (0.984 + 0.173i)6-s + (−0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s − 0.999·12-s + (−0.816 + 0.218i)13-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)18-s + (−0.866 − 0.5i)23-s + (0.984 − 0.173i)24-s + (−0.866 + 0.5i)25-s + (0.766 − 0.357i)26-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.939 − 0.342i)3-s + (0.939 − 0.342i)4-s + (0.984 + 0.173i)6-s + (−0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s − 0.999·12-s + (−0.816 + 0.218i)13-s + (0.766 − 0.642i)16-s + (−0.866 − 0.5i)18-s + (−0.866 − 0.5i)23-s + (0.984 − 0.173i)24-s + (−0.866 + 0.5i)25-s + (0.766 − 0.357i)26-s + (−0.500 − 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: $\chi_{3312} (1885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1602491063\)
\(L(\frac12)\) \(\approx\) \(0.1602491063\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
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 \( 1 + (0.816 - 0.218i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (0.424 - 1.58i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.96iT - T^{2} \)
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

−8.423777595171966715146854636367, −7.66521116230488656306949018304, −7.09207180230223307088438544802, −6.42770394316167809193112134240, −5.65044772626909303352756387359, −4.98684689193030925110569406966, −3.81723228164187809034795342102, −2.40106905707233089157570203865, −1.61979693058250565817763450030, −0.15298055314668360943750048170, 1.32703387755890142040701849437, 2.48392778703501194678969189999, 3.59389807466230502759949306531, 4.53275645507267321286356148295, 5.48714520541482334693527416997, 6.27385870333218916416470919619, 6.85089209321309664321133038945, 7.78063860980863483183632813672, 8.281589932970622926475974733686, 9.406292835202177244588694643767

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