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Properties

Label 2-520-104.101-c1-0-19
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: $\chi_{520} (101, \cdot )$
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

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

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