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Properties

Label 2-3312-3312.3035-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  + (0.342 + 0.939i)2-s + (−0.766 − 0.642i)3-s + (−0.766 + 0.642i)4-s + (0.342 − 0.939i)6-s + (−0.866 − 0.500i)8-s + (0.173 + 0.984i)9-s + 12-s + (−1.92 − 0.515i)13-s + (0.173 − 0.984i)16-s + (−0.866 + 0.499i)18-s + (0.866 − 0.5i)23-s + (0.342 + 0.939i)24-s + (0.866 + 0.5i)25-s + (−0.173 − 1.98i)26-s + (0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.766 − 0.642i)3-s + (−0.766 + 0.642i)4-s + (0.342 − 0.939i)6-s + (−0.866 − 0.500i)8-s + (0.173 + 0.984i)9-s + 12-s + (−1.92 − 0.515i)13-s + (0.173 − 0.984i)16-s + (−0.866 + 0.499i)18-s + (0.866 − 0.5i)23-s + (0.342 + 0.939i)24-s + (0.866 + 0.5i)25-s + (−0.173 − 1.98i)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: $\chi_{3312} (3035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6981784003\)
\(L(\frac12)\) \(\approx\) \(0.6981784003\)
\(L(1)\) 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 \( 1 + (0.766 + 0.642i)T \)
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 \( 1 + (1.92 + 0.515i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (0.816 - 0.218i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
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

−8.413468488277457175444552556541, −7.77206559086951260703958941680, −7.05653081676703815917550756778, −6.70417745401509375375453494500, −5.67868433339717993288979242825, −5.05191782342608247751538927081, −4.59521379544533738466424248582, −3.23596163724345787816025056863, −2.23562037993790805818573824593, −0.45914075670183646556579076121, 1.17982502413625647336649218138, 2.56063342114673745333201536434, 3.29454972442040000670568527980, 4.55795568525549950655376685508, 4.71698055235605581140749076862, 5.54135627566880886840224798392, 6.49853039405516828472261777911, 7.19618950688511753992511633874, 8.424738787718685759875862002149, 9.220353710504838952873352503714

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