Properties

Label 2-2520-5.4-c1-0-42
Degree $2$
Conductor $2520$
Sign $-0.970 + 0.241i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  + (2.17 − 0.539i)5-s + i·7-s − 5.41·11-s − 4.34i·13-s − 1.07i·17-s − 4.34·19-s + 6.34i·23-s + (4.41 − 2.34i)25-s − 8.83·29-s − 4.34·31-s + (0.539 + 2.17i)35-s − 8.68i·37-s − 8.34·41-s − 6.15i·43-s + 6.83i·47-s + ⋯
L(s)  = 1  + (0.970 − 0.241i)5-s + 0.377i·7-s − 1.63·11-s − 1.20i·13-s − 0.261i·17-s − 0.995·19-s + 1.32i·23-s + (0.883 − 0.468i)25-s − 1.64·29-s − 0.779·31-s + (0.0911 + 0.366i)35-s − 1.42i·37-s − 1.30·41-s − 0.938i·43-s + 0.997i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3564419069\)
\(L(\frac12)\) \(\approx\) \(0.3564419069\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2.17 + 0.539i)T \)
7 \( 1 - iT \)
good11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + 4.34iT - 13T^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 - 6.34iT - 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 + 4.34T + 31T^{2} \)
37 \( 1 + 8.68iT - 37T^{2} \)
41 \( 1 + 8.34T + 41T^{2} \)
43 \( 1 + 6.15iT - 43T^{2} \)
47 \( 1 - 6.83iT - 47T^{2} \)
53 \( 1 - 6.18iT - 53T^{2} \)
59 \( 1 + 6.83T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 0.680T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 - 6.49T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−8.602884350599730264582457257345, −7.78384615566134339167288637722, −7.17717200424019741746383155146, −5.78608212811687808460838916957, −5.63948463757538362062992275367, −4.87787988195624605664006181936, −3.52452300524485334027501002105, −2.58011024298712193314680025982, −1.78946046660152835524318613624, −0.10272849062823947252017543617, 1.77049294945853635274867955725, 2.42363650973790615175832615393, 3.55592530133416368806222763975, 4.68581386536221207652273383825, 5.28931250826951267184726123926, 6.33986870572924351825375377067, 6.77285867514921138156500711782, 7.77328341163687838646927342708, 8.509826673041543902938927368016, 9.320242712289931058559898333498

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