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Properties

Label 2-3332-476.123-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.710 - 0.704i$
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 − 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 + (−1.36 − 0.366i)5-s − 0.999i·8-s + (−0.866 + 0.5i)9-s + (−1.36 + 0.366i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.710 - 0.704i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.710 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1898708935\)
\(L(\frac12)\) \(\approx\) \(0.1898708935\)
\(L(1)\) 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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.289161649645565420707567005761, −7.36562339920749780983907748765, −7.10165930573071625863095662136, −5.64849999007346632549395337950, −5.21035404823262886925983477800, −4.45418374612990984715044325718, −3.59763740073535095213674976767, −2.89198420898493790125070914590, −1.81961376715216428207290547883, −0.07646915520612395113461510800, 2.35716060353809213501587338268, 3.21686003581738107936935137237, 3.73938351189108354983259846442, 4.81782918993239323969554690927, 5.27939154624651060754916908468, 6.36398225657065211474226224729, 7.06509361887206318236065634109, 7.75198788706242349682453386726, 8.052993469649471103814121912231, 9.125636392359144429515718338591

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