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Properties

Label 2-520-104.69-c1-0-44
Degree $2$
Conductor $520$
Sign $0.351 + 0.936i$
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  + (1.41 + 0.0204i)2-s + (−1.50 − 0.869i)3-s + (1.99 + 0.0579i)4-s − 5-s + (−2.11 − 1.25i)6-s + (0.987 − 0.570i)7-s + (2.82 + 0.122i)8-s + (0.0109 + 0.0189i)9-s + (−1.41 − 0.0204i)10-s + (1.00 − 1.73i)11-s + (−2.95 − 1.82i)12-s + (0.482 − 3.57i)13-s + (1.40 − 0.785i)14-s + (1.50 + 0.869i)15-s + (3.99 + 0.231i)16-s + (1.50 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0144i)2-s + (−0.869 − 0.501i)3-s + (0.999 + 0.0289i)4-s − 0.447·5-s + (−0.861 − 0.514i)6-s + (0.373 − 0.215i)7-s + (0.999 + 0.0434i)8-s + (0.00365 + 0.00633i)9-s + (−0.447 − 0.00648i)10-s + (0.301 − 0.522i)11-s + (−0.854 − 0.526i)12-s + (0.133 − 0.990i)13-s + (0.376 − 0.210i)14-s + (0.388 + 0.224i)15-s + (0.998 + 0.0579i)16-s + (0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.351 + 0.936i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59259 - 1.10381i\)
\(L(\frac12)\) \(\approx\) \(1.59259 - 1.10381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0204i)T \)
5 \( 1 + T \)
13 \( 1 + (-0.482 + 3.57i)T \)
good3 \( 1 + (1.50 + 0.869i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.987 + 0.570i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.00 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.50 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.66 + 4.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 + 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.94iT - 31T^{2} \)
37 \( 1 + (4.33 - 7.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.975 - 0.563i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.66 + 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.99iT - 47T^{2} \)
53 \( 1 - 6.13iT - 53T^{2} \)
59 \( 1 + (-1.62 - 2.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.54 - 0.892i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.15 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.89 + 2.82i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.8iT - 73T^{2} \)
79 \( 1 - 5.25T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (1.76 + 1.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.354 - 0.204i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05278801846486837064384362293, −10.35146880405272444997197990605, −8.667598486195741759984972140689, −7.72708446560210740161771342888, −6.74732483164879632513670134760, −6.03665972830782231904516145036, −5.13458256638385869095384000735, −4.06170087812801651323987718697, −2.85856393840169570163738218745, −0.997444900436647094914910921975, 1.88850916033401435460695271758, 3.55106948883912533457875332637, 4.56100399679729024287730065246, 5.20547534146198696864356104170, 6.22710314253869711528069030697, 7.13776284564182424171928469434, 8.171387779773613522349872629257, 9.501178339743183115536191514299, 10.61178295639725319990275438190, 11.13806426092183192811025495136

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