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Properties

Label 2-3332-476.283-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.686 + 0.727i$
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  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.389 + 0.0255i)5-s + (−0.382 − 0.923i)8-s + (−0.793 − 0.608i)9-s + (0.0255 + 0.389i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (−0.793 + 0.608i)17-s + (0.499 − 0.866i)18-s + (−0.382 + 0.0761i)20-s + (−0.840 − 0.110i)25-s + (0.860 − 1.12i)26-s + (1.08 − 1.63i)29-s + (0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.389 + 0.0255i)5-s + (−0.382 − 0.923i)8-s + (−0.793 − 0.608i)9-s + (0.0255 + 0.389i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (−0.793 + 0.608i)17-s + (0.499 − 0.866i)18-s + (−0.382 + 0.0761i)20-s + (−0.840 − 0.110i)25-s + (0.860 − 1.12i)26-s + (1.08 − 1.63i)29-s + (0.608 + 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6653931281\)
\(L(\frac12)\) \(\approx\) \(0.6653931281\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
17 \( 1 + (0.793 - 0.608i)T \)
good3 \( 1 + (0.793 + 0.608i)T^{2} \)
5 \( 1 + (-0.389 - 0.0255i)T + (0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.130 + 0.991i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (-0.793 + 0.608i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.793 - 0.608i)T^{2} \)
37 \( 1 + (-0.293 + 0.257i)T + (0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.996 + 0.491i)T + (0.608 + 0.793i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.867 + 1.75i)T + (-0.608 + 0.793i)T^{2} \)
79 \( 1 + (0.793 - 0.608i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
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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

−8.550259950371433316876627303962, −7.972400102603669788702417507724, −7.20516161010512441728366664738, −6.27926518776575093251406471256, −5.89307199127425085729224953858, −5.07576152964936403048344056932, −4.22410155083978792882552183266, −3.28186077696789439818565916654, −2.28198360652221872361984502313, −0.36441365935378496350718591832, 1.55114393203067532580473553242, 2.45762128288336573550504487210, 3.07617604670919702897720831915, 4.37661882790309485208598824299, 4.87914889574865329692016516134, 5.64466620696335795779520409321, 6.60170355840747367616998430835, 7.48324240417657348053707318135, 8.524301034805297344984357047931, 8.936301806673843055603391599677

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