Properties

Label 2-4560-76.75-c1-0-31
Degree $2$
Conductor $4560$
Sign $0.937 - 0.349i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 5.16i·7-s + 9-s + 4.48i·11-s + 1.15i·13-s − 15-s + 0.995·17-s + (3.36 + 2.77i)19-s − 5.16i·21-s + 2.60i·23-s + 25-s + 27-s + 9.93i·29-s + 2.14·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.95i·7-s + 0.333·9-s + 1.35i·11-s + 0.319i·13-s − 0.258·15-s + 0.241·17-s + (0.770 + 0.637i)19-s − 1.12i·21-s + 0.542i·23-s + 0.200·25-s + 0.192·27-s + 1.84i·29-s + 0.385·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 0.937 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.140943868\)
\(L(\frac12)\) \(\approx\) \(2.140943868\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 + (-3.36 - 2.77i)T \)
good7 \( 1 + 5.16iT - 7T^{2} \)
11 \( 1 - 4.48iT - 11T^{2} \)
13 \( 1 - 1.15iT - 13T^{2} \)
17 \( 1 - 0.995T + 17T^{2} \)
23 \( 1 - 2.60iT - 23T^{2} \)
29 \( 1 - 9.93iT - 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 + 7.02iT - 37T^{2} \)
41 \( 1 - 11.4iT - 41T^{2} \)
43 \( 1 - 3.55iT - 43T^{2} \)
47 \( 1 + 5.00iT - 47T^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 - 4.77T + 59T^{2} \)
61 \( 1 + 2.92T + 61T^{2} \)
67 \( 1 - 8.46T + 67T^{2} \)
71 \( 1 + 6.99T + 71T^{2} \)
73 \( 1 - 3.86T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + 15.9iT - 97T^{2} \)
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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

−8.107654560409285007296308120052, −7.57309477673236485114981850807, −7.13442210877204424058154215067, −6.55761002903052828014196384435, −5.13149350976694526274392363926, −4.50788042425906579548644475755, −3.76954874744139196009541554644, −3.24518908493037919369352915500, −1.82412695720394957815289226249, −1.00958915331023038534414514177, 0.65453354341484917337886276538, 2.17688773866185915701699143212, 2.85329469955128846841478590466, 3.45399329544805269992229774555, 4.58183185210884014583464211954, 5.49058539777091259844223134237, 5.95802508672475329778480014282, 6.79642649645390023307830683421, 7.991594662138008749458788251660, 8.187590113262421318020434635528

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