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Properties

Label 2-520-13.4-c1-0-11
Degree $2$
Conductor $520$
Sign $-0.0423 + 0.999i$
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  + (1.43 − 2.48i)3-s + i·5-s + (0.0113 − 0.00658i)7-s + (−2.62 − 4.53i)9-s + (0.541 + 0.312i)11-s + (1.97 − 3.01i)13-s + (2.48 + 1.43i)15-s + (−3.78 − 6.54i)17-s + (6.29 − 3.63i)19-s − 0.0377i·21-s + (−1.22 + 2.13i)23-s − 25-s − 6.43·27-s + (−1.15 + 2.00i)29-s + 8.02i·31-s + ⋯
L(s)  = 1  + (0.828 − 1.43i)3-s + 0.447i·5-s + (0.00430 − 0.00248i)7-s + (−0.873 − 1.51i)9-s + (0.163 + 0.0942i)11-s + (0.547 − 0.837i)13-s + (0.641 + 0.370i)15-s + (−0.916 − 1.58i)17-s + (1.44 − 0.833i)19-s − 0.00824i·21-s + (−0.256 + 0.444i)23-s − 0.200·25-s − 1.23·27-s + (−0.215 + 0.373i)29-s + 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0423 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0423 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.0423 + 0.999i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.0423 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27476 - 1.32997i\)
\(L(\frac12)\) \(\approx\) \(1.27476 - 1.32997i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-1.97 + 3.01i)T \)
good3 \( 1 + (-1.43 + 2.48i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.0113 + 0.00658i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.541 - 0.312i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.78 + 6.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.29 + 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.22 - 2.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.15 - 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.02iT - 31T^{2} \)
37 \( 1 + (3.65 + 2.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 1.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.39 - 2.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.00iT - 47T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 + (11.6 - 6.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.28 - 5.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.81 - 3.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.2 + 7.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.98iT - 73T^{2} \)
79 \( 1 - 2.02T + 79T^{2} \)
83 \( 1 + 2.35iT - 83T^{2} \)
89 \( 1 + (-4.50 - 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.02 + 4.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.81637136022381080880626060398, −9.441979519967381795353415795180, −8.824783298525500995087163135919, −7.67287347118950559851900630065, −7.22634579593162771291411724651, −6.37629698877426067637402314345, −5.10636010537989316240223784707, −3.30438893654510820993026081147, −2.58287055536551451700745782435, −1.08947827079650667427459457065, 2.03161299245313948593089133296, 3.67139889096209086959803556391, 4.10017184155106943706403941508, 5.26274782730777445691905665210, 6.36942952710762430213058178919, 7.938164288257196839904487461016, 8.601220470102421697796618355305, 9.360182856364490902857829348782, 10.02258214424741167060964178913, 10.91608111714844983597290795421

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