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Properties

Label 2-3332-476.387-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.803 - 0.595i$
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 + (−1.36 + 0.366i)5-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)10-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.999 − 0.999i)20-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.866 − 0.499i)32-s + 0.999i·34-s − 0.999i·36-s + (1.36 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.36 + 0.366i)5-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (1.36 + 0.366i)10-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.999 − 0.999i)20-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.866 − 0.499i)32-s + 0.999i·34-s − 0.999i·36-s + (1.36 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4230464141\)
\(L(\frac12)\) \(\approx\) \(0.4230464141\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + T^{2} \)
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 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−8.747501822276882638344351859079, −8.309394413620756412203976266732, −7.51366325706378503990873009210, −6.93425905449879630512231962839, −6.17715472083869137951955940554, −4.85075914889955199496866068541, −3.94578985778106671440609392825, −3.16736292453920796230399277180, −2.55430590685711043479117431061, −0.893142184475783003192371305848, 0.45204804704840273107825280082, 1.97040737002892401057212561163, 3.07668220427799491294578385916, 4.22355377291641730510013127029, 4.94478206096547218130774118951, 5.90580615144908169331305987135, 6.59912875089310583426591640024, 7.53814292094373224949177987556, 8.135009772078663439128528463017, 8.450008617808554194526420439779

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