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Properties

Label 2-3312-207.22-c0-0-2
Degree $2$
Conductor $3312$
Sign $-0.5 + 0.866i$
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  + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.300i)13-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)27-s + (−0.766 − 1.32i)29-s + (0.766 − 1.32i)31-s + (0.326 + 0.118i)39-s + (0.939 − 1.62i)41-s + (−0.939 − 1.62i)47-s + (−0.5 + 0.866i)49-s + (−0.5 + 0.866i)59-s + (−0.939 − 0.342i)69-s − 1.53·71-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.300i)13-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)27-s + (−0.766 − 1.32i)29-s + (0.766 − 1.32i)31-s + (0.326 + 0.118i)39-s + (0.939 − 1.62i)41-s + (−0.939 − 1.62i)47-s + (−0.5 + 0.866i)49-s + (−0.5 + 0.866i)59-s + (−0.939 − 0.342i)69-s − 1.53·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9319233919\)
\(L(\frac12)\) \(\approx\) \(0.9319233919\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−8.402737842285774031334012264490, −7.77360969976815474106189445864, −7.12057519508047020075148337135, −6.27594500939649707537006727473, −5.81192451635427850631623256492, −4.76141666439504725237647968640, −3.89637088662067014838544300489, −2.62427628479996310591380233318, −2.01077290908997860599356854043, −0.57620412015851202555279693057, 1.46512663811570902440759519743, 3.00024021823520945383790710754, 3.45312113410931377753015273528, 4.58631222271144710655455521927, 5.14191427390223840336679624345, 5.89921996395494798537470069434, 6.75496338731418011143061300254, 7.66992473307208749715301303963, 8.405042288392096504321186511081, 9.285055124769421921189398284267

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