Properties

Label 2-520-13.3-c1-0-2
Degree $2$
Conductor $520$
Sign $-0.859 - 0.511i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
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  + (1.5 + 2.59i)3-s − 5-s + (−1.5 + 2.59i)7-s + (−3 + 5.19i)9-s + (−2.5 − 4.33i)11-s + (−1 + 3.46i)13-s + (−1.5 − 2.59i)15-s + (−1.5 + 2.59i)17-s + (2.5 − 4.33i)19-s − 9·21-s + (1.5 + 2.59i)23-s + 25-s − 9·27-s + (2.5 + 4.33i)29-s + 8·31-s + ⋯
L(s)  = 1  + (0.866 + 1.49i)3-s − 0.447·5-s + (−0.566 + 0.981i)7-s + (−1 + 1.73i)9-s + (−0.753 − 1.30i)11-s + (−0.277 + 0.960i)13-s + (−0.387 − 0.670i)15-s + (−0.363 + 0.630i)17-s + (0.573 − 0.993i)19-s − 1.96·21-s + (0.312 + 0.541i)23-s + 0.200·25-s − 1.73·27-s + (0.464 + 0.804i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.358515 + 1.30448i\)
\(L(\frac12)\) \(\approx\) \(0.358515 + 1.30448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15773783095593600931307529550, −10.18449438595581796237140610088, −9.391744724807741738305258272243, −8.686318405517961695620533635450, −8.161226599088127856961603074011, −6.60836040231302759417409546644, −5.36376423050988807332413938945, −4.51991206602175920224873945799, −3.29211140995584102269353189495, −2.71583420946706239258448939980, 0.70949886755812806530210872213, 2.33609814881258665999937591805, 3.29854745563180318756575379034, 4.67760186966638720214880640810, 6.26902721788690842126075635335, 7.20104346398524019225269104052, 7.65812878364927031632105439058, 8.315698764866817697863712539006, 9.679953984654399956491786010887, 10.30307116726420003129175886689

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