Properties

Label 2-1620-180.167-c0-0-5
Degree $2$
Conductor $1620$
Sign $0.979 - 0.203i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
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 + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.133 − 0.5i)13-s + (0.500 + 0.866i)16-s + (−1.22 + 1.22i)17-s + (0.965 − 0.258i)20-s − 1.00i·25-s − 0.517i·26-s + (−0.965 − 1.67i)29-s + (0.258 + 0.965i)32-s + (−1.49 + 0.866i)34-s + (1.36 + 1.36i)37-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.133 − 0.5i)13-s + (0.500 + 0.866i)16-s + (−1.22 + 1.22i)17-s + (0.965 − 0.258i)20-s − 1.00i·25-s − 0.517i·26-s + (−0.965 − 1.67i)29-s + (0.258 + 0.965i)32-s + (−1.49 + 0.866i)34-s + (1.36 + 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.203874691\)
\(L(\frac12)\) \(\approx\) \(2.203874691\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 - 0.517T + T^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.704561092003897085859492748041, −8.550814394519954855215459324942, −8.110674161046732775698014627719, −6.97309417819355674225366092139, −6.13721332240979299709863113527, −5.60420743905689515757235479182, −4.61893406479561407886260633076, −3.96194431360034616461448886154, −2.63780158461373181775080698332, −1.71537300918316557586658568835, 1.76460930281198264879282382809, 2.62495250444300182537504969738, 3.52442965882191482583744149730, 4.65723356416275828324993994770, 5.35516832768796322301801023310, 6.35464189172344624058535781347, 6.88291827769189969699967262845, 7.61035683272436962632606430126, 9.106448107606240040960735728627, 9.585781410086267703735639241071

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