Properties

Label 2-2520-7.4-c1-0-28
Degree $2$
Conductor $2520$
Sign $-0.283 + 0.958i$
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  + (−0.5 − 0.866i)5-s + (−0.780 − 2.52i)7-s + (−2.39 + 4.15i)11-s + 5.94·13-s + (3.41 − 5.91i)17-s + (−4.26 − 7.39i)19-s + (4.48 + 7.77i)23-s + (−0.499 + 0.866i)25-s + 0.414·29-s + (−2.80 + 4.86i)31-s + (−1.79 + 1.93i)35-s + (−2.79 − 4.84i)37-s + 0.325·41-s + 3.90·43-s + (−4.86 − 8.43i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.294 − 0.955i)7-s + (−0.722 + 1.25i)11-s + 1.64·13-s + (0.828 − 1.43i)17-s + (−0.979 − 1.69i)19-s + (0.936 + 1.62i)23-s + (−0.0999 + 0.173i)25-s + 0.0770·29-s + (−0.504 + 0.873i)31-s + (−0.304 + 0.327i)35-s + (−0.460 − 0.797i)37-s + 0.0508·41-s + 0.595·43-s + (−0.710 − 1.23i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.283 + 0.958i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ -0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.315177691\)
\(L(\frac12)\) \(\approx\) \(1.315177691\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.780 + 2.52i)T \)
good11 \( 1 + (2.39 - 4.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.94T + 13T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.26 + 7.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.48 - 7.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.414T + 29T^{2} \)
31 \( 1 + (2.80 - 4.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.79 + 4.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.325T + 41T^{2} \)
43 \( 1 - 3.90T + 43T^{2} \)
47 \( 1 + (4.86 + 8.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.927 - 1.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.58 + 7.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.33 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 + 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 + (-4.40 + 7.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.81 + 3.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + (0.646 + 1.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.796220689437342980184644365494, −7.75768271705594307567651561907, −7.18734221372885859163542771443, −6.62258694518494790899406761935, −5.30014236757050083613570958557, −4.82798380875791603603968804871, −3.80675782324348140685379173611, −3.03950519460009906898456087830, −1.64678374875792205053776119784, −0.47049242538506832725394069240, 1.29442023127373395396890083953, 2.60219077953659596950827113074, 3.44024078922326161992180072146, 4.12406512887157212443094084307, 5.61458981820965478720406970815, 5.99532369342487584752079417167, 6.52528598779313585901472999555, 7.975090327529401159552333045976, 8.383865206158017291381923143615, 8.791783626095594299222209117705

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