Properties

Label 2-4560-76.75-c1-0-3
Degree $2$
Conductor $4560$
Sign $-0.743 - 0.668i$
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 + 0.549i·7-s + 9-s − 1.98i·11-s − 2.38i·13-s − 15-s − 4.49·17-s + (−4.14 + 1.35i)19-s + 0.549i·21-s − 2.88i·23-s + 25-s + 27-s + 3.54i·29-s + 0.0281·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.207i·7-s + 0.333·9-s − 0.598i·11-s − 0.662i·13-s − 0.258·15-s − 1.09·17-s + (−0.950 + 0.309i)19-s + 0.119i·21-s − 0.602i·23-s + 0.200·25-s + 0.192·27-s + 0.658i·29-s + 0.00505·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.743 - 0.668i$
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: \((2,\ 4560,\ (\ :1/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5788805788\)
\(L(\frac12)\) \(\approx\) \(0.5788805788\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.14 - 1.35i)T \)
good7 \( 1 - 0.549iT - 7T^{2} \)
11 \( 1 + 1.98iT - 11T^{2} \)
13 \( 1 + 2.38iT - 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
23 \( 1 + 2.88iT - 23T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 - 0.0281T + 31T^{2} \)
37 \( 1 + 3.37iT - 37T^{2} \)
41 \( 1 - 4.40iT - 41T^{2} \)
43 \( 1 - 6.83iT - 43T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 - 3.79T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 4.92T + 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 + 7.76T + 73T^{2} \)
79 \( 1 - 2.04T + 79T^{2} \)
83 \( 1 - 7.95iT - 83T^{2} \)
89 \( 1 + 0.0323iT - 89T^{2} \)
97 \( 1 - 0.596iT - 97T^{2} \)
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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

−8.745309844930971811526272161849, −7.913159395211633241980874935754, −7.34445428615974462763728141716, −6.37276688778603609968535352013, −5.84152852768893272209529869280, −4.64177702996062602301861536585, −4.17204698186785707073239197493, −3.09842441499079502478294393530, −2.53739950022576723538010145523, −1.28723816146179000988413952599, 0.14237984690237156538186712112, 1.76031649134197587338873866781, 2.42261232832788478686943091094, 3.57893495439844407489530399044, 4.24331137083147362457439237933, 4.81540033445201857559474869108, 5.94584999527185613860983449925, 6.93812829295592863589190212138, 7.15710279980435331858686679926, 8.138135961181193451734442689206

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