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Properties

Label 2-2520-105.104-c1-0-22
Degree $2$
Conductor $2520$
Sign $0.969 - 0.245i$
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  + (−1.52 + 1.63i)5-s + (−0.114 − 2.64i)7-s + 3.82i·11-s + 6.36·13-s + 0.444i·17-s − 4.39i·19-s − 4.65·23-s + (−0.345 − 4.98i)25-s − 9.53i·29-s + 4.88i·31-s + (4.49 + 3.84i)35-s + 5.61i·37-s + 8.01·41-s − 3.33i·43-s + 12.8i·47-s + ⋯
L(s)  = 1  + (−0.682 + 0.731i)5-s + (−0.0433 − 0.999i)7-s + 1.15i·11-s + 1.76·13-s + 0.107i·17-s − 1.00i·19-s − 0.971·23-s + (−0.0691 − 0.997i)25-s − 1.77i·29-s + 0.877i·31-s + (0.760 + 0.649i)35-s + 0.922i·37-s + 1.25·41-s − 0.508i·43-s + 1.87i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.969 - 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604440420\)
\(L(\frac12)\) \(\approx\) \(1.604440420\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 - 1.63i)T \)
7 \( 1 + (0.114 + 2.64i)T \)
good11 \( 1 - 3.82iT - 11T^{2} \)
13 \( 1 - 6.36T + 13T^{2} \)
17 \( 1 - 0.444iT - 17T^{2} \)
19 \( 1 + 4.39iT - 19T^{2} \)
23 \( 1 + 4.65T + 23T^{2} \)
29 \( 1 + 9.53iT - 29T^{2} \)
31 \( 1 - 4.88iT - 31T^{2} \)
37 \( 1 - 5.61iT - 37T^{2} \)
41 \( 1 - 8.01T + 41T^{2} \)
43 \( 1 + 3.33iT - 43T^{2} \)
47 \( 1 - 12.8iT - 47T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 - 5.81T + 59T^{2} \)
61 \( 1 + 5.28iT - 61T^{2} \)
67 \( 1 - 9.42iT - 67T^{2} \)
71 \( 1 + 2.35iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 4.80T + 97T^{2} \)
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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

−8.886720314245766690724905287885, −7.920428669696890036429460156968, −7.54919548271243170136395712863, −6.56881346578210396851603024351, −6.20259352713139883742611343780, −4.70650219616234333521214871534, −4.08591928004016236994437481403, −3.40405107109126646327914251763, −2.21594488461547262987091600271, −0.842223660021617508209690825748, 0.795372286826283523565978435092, 1.97869195769285722791987824375, 3.47542244848138697357571205196, 3.77825617273190675314715428148, 5.07759001692832469165096421603, 5.83836065306720622963824961253, 6.27515641605585872906817737844, 7.61287238726659363461799826451, 8.322462238034084888041601720125, 8.735447715454359441326603551493

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