Properties

Label 2-4560-76.75-c1-0-47
Degree $2$
Conductor $4560$
Sign $0.0454 + 0.998i$
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 − 2.60i·7-s + 9-s + 1.74i·11-s − 2.60i·13-s + 15-s − 0.802·17-s + (0.197 + 4.35i)19-s + 2.60i·21-s − 1.23i·23-s + 25-s − 27-s + 1.74i·29-s + 8.15·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.985i·7-s + 0.333·9-s + 0.526i·11-s − 0.723i·13-s + 0.258·15-s − 0.194·17-s + (0.0454 + 0.998i)19-s + 0.569i·21-s − 0.256i·23-s + 0.200·25-s − 0.192·27-s + 0.324i·29-s + 1.46·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0454 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.0454 + 0.998i$
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: \((2,\ 4560,\ (\ :1/2),\ 0.0454 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.077512382\)
\(L(\frac12)\) \(\approx\) \(1.077512382\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.197 - 4.35i)T \)
good7 \( 1 + 2.60iT - 7T^{2} \)
11 \( 1 - 1.74iT - 11T^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
17 \( 1 + 0.802T + 17T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 - 1.74iT - 29T^{2} \)
31 \( 1 - 8.15T + 31T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + 6.96iT - 41T^{2} \)
43 \( 1 + 2.97iT - 43T^{2} \)
47 \( 1 - 9.93iT - 47T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 - 1.60T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 9.35T + 67T^{2} \)
71 \( 1 - 1.60T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 4.72iT - 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + 8.85iT - 97T^{2} \)
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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

−7.956036394535218266590007544820, −7.43563485698521933764535788444, −6.77439042233843876745341073606, −5.98195070349547571245932593884, −5.16757008533308605626568085436, −4.32795370068358497285731687450, −3.81107837955976643731272972020, −2.73542969040162127143386050123, −1.42754528244400970001071374784, −0.42004704081597296589588180401, 0.926937810301434658420741285502, 2.23215539739749270541178352634, 3.05746133603291805649396212742, 4.12434096134390871778137292017, 4.86110298234246375981173132607, 5.53776747292727632558226498954, 6.43042844497445137694550016029, 6.81035091562320710248921562521, 7.87196483521485702807652076299, 8.501131373961558705722737304224

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