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Properties

Label 2-3332-476.135-c0-0-9
Degree $2$
Conductor $3332$
Sign $-0.126 + 0.991i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.73 − i)5-s + 0.999·8-s + (0.5 − 0.866i)9-s + (1.73 − 0.999i)10-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.499 + 0.866i)18-s + 1.99i·20-s + (1.49 + 2.59i)25-s + (−0.499 − 0.866i)32-s + 0.999i·34-s − 0.999·36-s + (−1.73 − i)40-s − 2i·41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.73 − i)5-s + 0.999·8-s + (0.5 − 0.866i)9-s + (1.73 − 0.999i)10-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.499 + 0.866i)18-s + 1.99i·20-s + (1.49 + 2.59i)25-s + (−0.499 − 0.866i)32-s + 0.999i·34-s − 0.999·36-s + (−1.73 − i)40-s − 2i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.126 + 0.991i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4618346876\)
\(L(\frac12)\) \(\approx\) \(0.4618346876\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.532967724745047667391441591565, −7.79523780276318624641051790341, −7.33862042633079849880418257397, −6.62419169917635739157837886597, −5.51709714189326215017318102251, −4.83604101082938134552442001851, −4.06928304415945838505622781405, −3.40545964801301604025969064137, −1.38505737588225704126891175396, −0.38418984063941213275853271134, 1.39964661696313612361359169203, 2.77834836456746713932502168881, 3.28399196743587753970635680930, 4.25613958178354056521224331237, 4.67813307554811207406880013659, 6.17372458509237502090917554475, 7.25330922889510287153655716873, 7.66940866538205969794832502131, 8.103163413473830652361932240069, 8.932144562634432912288354534455

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