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Properties

Label 2-3332-476.319-c0-0-0
Degree $2$
Conductor $3332$
Sign $0.510 - 0.859i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.366 + 1.36i)5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.366 − 1.36i)10-s + 2·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.499 + 0.866i)18-s + (0.999 + 0.999i)20-s + (−0.866 − 0.5i)25-s + (−1.73 + i)26-s + (−1 − i)29-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.366 + 1.36i)5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.366 − 1.36i)10-s + 2·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.499 + 0.866i)18-s + (0.999 + 0.999i)20-s + (−0.866 − 0.5i)25-s + (−1.73 + i)26-s + (−1 − i)29-s + (0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.510 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9685260811\)
\(L(\frac12)\) \(\approx\) \(0.9685260811\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
17 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 - 2T + T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.005703529698859567353863122746, −7.87628332121347762491039638003, −7.64051642119053652538365908114, −6.68487333782449276333389837327, −6.23298662726150483540466471606, −5.57187666913145199100515301077, −3.99545074357470701238648966509, −3.52462170581088793703702568455, −2.25003092462770182534077476561, −1.11293147012693427219868294174, 1.11758881126011125926257269677, 1.53683453102130554480168920674, 3.09752956401576953025995233556, 3.94570576014562021581925710889, 4.64947582485251837650653996045, 5.65296988350534869298472054807, 6.61130599457158151863301787413, 7.58101760149600110564752394401, 8.099857452273330144927158153607, 8.696523585368969429833165058444

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