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Properties

Label 2-3332-476.179-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.142 + 0.989i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−1.83 + 0.241i)5-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (1.83 + 0.241i)10-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.499 + 0.866i)18-s + (−1.70 − 0.707i)20-s + (2.33 − 0.624i)25-s + (−0.707 + 1.70i)29-s + (−0.258 − 0.965i)32-s − i·34-s + (0.707 − 0.707i)36-s + (−0.241 − 1.83i)37-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−1.83 + 0.241i)5-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (1.83 + 0.241i)10-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.499 + 0.866i)18-s + (−1.70 − 0.707i)20-s + (2.33 − 0.624i)25-s + (−0.707 + 1.70i)29-s + (−0.258 − 0.965i)32-s − i·34-s + (0.707 − 0.707i)36-s + (−0.241 − 1.83i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.142 + 0.989i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.142 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3780176981\)
\(L(\frac12)\) \(\approx\) \(0.3780176981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 \)
17 \( 1 + (-0.258 - 0.965i)T \)
good3 \( 1 + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T^{2} \)
29 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)T^{2} \)
41 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (1.12 + 1.46i)T + (-0.258 + 0.965i)T^{2} \)
79 \( 1 + (-0.258 - 0.965i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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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

−8.807915988141060156976729614634, −7.82761073893858858827806033756, −7.30860066372174603237912866283, −6.78069591179218972629712007319, −5.80542508151205201063954352372, −4.42053803996716639669785761405, −3.53494915371445483046303017657, −3.30949861999462570812819989718, −1.72835739662762071265245408365, −0.38077843168179938311252986372, 1.05328748996031345716370664575, 2.46550938726616524475874508777, 3.41771687755488308128885370052, 4.51682642026728674104845192389, 5.09621884885345239523807194106, 6.27069439053920585203027812680, 7.16674662065826267922854708039, 7.77028542320002903380815931567, 8.017007572449845541935661207879, 8.820326321199279934147053790763

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