Properties

Label 2-4560-76.75-c1-0-39
Degree $2$
Conductor $4560$
Sign $0.999 - 0.0171i$
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 + 1.25i·7-s + 9-s + 0.630i·11-s + 3.96i·13-s − 15-s − 6.90·17-s + (2.11 − 3.81i)19-s − 1.25i·21-s − 6.49i·23-s + 25-s − 27-s − 2.83i·29-s + 7.48·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.473i·7-s + 0.333·9-s + 0.189i·11-s + 1.09i·13-s − 0.258·15-s − 1.67·17-s + (0.485 − 0.874i)19-s − 0.273i·21-s − 1.35i·23-s + 0.200·25-s − 0.192·27-s − 0.526i·29-s + 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 - 0.0171i$
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: \((2,\ 4560,\ (\ :1/2),\ 0.999 - 0.0171i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.539796556\)
\(L(\frac12)\) \(\approx\) \(1.539796556\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 + 3.81i)T \)
good7 \( 1 - 1.25iT - 7T^{2} \)
11 \( 1 - 0.630iT - 11T^{2} \)
13 \( 1 - 3.96iT - 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
23 \( 1 + 6.49iT - 23T^{2} \)
29 \( 1 + 2.83iT - 29T^{2} \)
31 \( 1 - 7.48T + 31T^{2} \)
37 \( 1 + 9.57iT - 37T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 + 7.37iT - 43T^{2} \)
47 \( 1 + 3.33iT - 47T^{2} \)
53 \( 1 - 9.95iT - 53T^{2} \)
59 \( 1 - 9.92T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 7.82T + 67T^{2} \)
71 \( 1 + 1.41T + 71T^{2} \)
73 \( 1 - 7.64T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 - 9.45iT - 83T^{2} \)
89 \( 1 - 4.52iT - 89T^{2} \)
97 \( 1 + 3.90iT - 97T^{2} \)
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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

−8.535360506338431587824758427095, −7.43651221636249336650937061446, −6.56175813565123375399988822031, −6.41795808464730972999552960424, −5.35525042499693529333910249899, −4.61725265701313061030402212483, −4.08353145136362992633932332640, −2.55486135917603424188996782986, −2.11709967254400859871954123100, −0.67147440344093626947873014202, 0.75478949611862918045126438782, 1.77723365497177063603607282720, 2.94637042352749046771741504652, 3.79389942318803830435738747078, 4.76992045095052666943212253306, 5.37281964820982210188537058197, 6.17991797577030426889883239685, 6.73075379523426691814363235877, 7.59881733329374227678668296868, 8.234557985534611547263375415444

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