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Properties

Label 2-4560-76.75-c1-0-7
Degree $2$
Conductor $4560$
Sign $-0.514 - 0.857i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.514 - 0.857i$
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: \((2,\ 4560,\ (\ :1/2),\ -0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.279711324\)
\(L(\frac12)\) \(\approx\) \(1.279711324\)
\(L(\frac{3}{2})\) 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

−8.796711855572982202380229922494, −7.68139977944554123595197009654, −7.31507567065361015864663089799, −6.32653902220003963004255533719, −5.86232119764833020642360233553, −4.56874017859424683872854191105, −4.15898814484020537493915832490, −3.21062029883110436819989860251, −2.12720914766792318351834810417, −1.51190529404512094523002961167, 0.28758736628474856367518198037, 1.88351526168905699059237923634, 2.50037576391664910341053757270, 3.32150454703269618248609350079, 4.41653721920635959776231599918, 5.03963013501031381522950695046, 5.92876785380651944202396621105, 6.73396985435113969745607953531, 7.26546837933457955465993487624, 8.345517555605076558712701427206

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