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Properties

Label 2-520-104.101-c1-0-18
Degree $2$
Conductor $520$
Sign $-0.850 - 0.526i$
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  + (0.449 + 1.34i)2-s + (−0.700 + 0.404i)3-s + (−1.59 + 1.20i)4-s + 5-s + (−0.856 − 0.757i)6-s + (3.38 + 1.95i)7-s + (−2.33 − 1.59i)8-s + (−1.17 + 2.03i)9-s + (0.449 + 1.34i)10-s + (0.223 + 0.387i)11-s + (0.630 − 1.48i)12-s + (3.01 − 1.98i)13-s + (−1.09 + 5.41i)14-s + (−0.700 + 0.404i)15-s + (1.09 − 3.84i)16-s + (−4.00 + 6.94i)17-s + ⋯
L(s)  = 1  + (0.317 + 0.948i)2-s + (−0.404 + 0.233i)3-s + (−0.798 + 0.602i)4-s + 0.447·5-s + (−0.349 − 0.309i)6-s + (1.27 + 0.737i)7-s + (−0.824 − 0.565i)8-s + (−0.390 + 0.677i)9-s + (0.142 + 0.424i)10-s + (0.0674 + 0.116i)11-s + (0.182 − 0.429i)12-s + (0.835 − 0.550i)13-s + (−0.293 + 1.44i)14-s + (−0.180 + 0.104i)15-s + (0.273 − 0.961i)16-s + (−0.972 + 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.850 - 0.526i$
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: \((2,\ 520,\ (\ :1/2),\ -0.850 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406268 + 1.42736i\)
\(L(\frac12)\) \(\approx\) \(0.406268 + 1.42736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.449 - 1.34i)T \)
5 \( 1 - T \)
13 \( 1 + (-3.01 + 1.98i)T \)
good3 \( 1 + (0.700 - 0.404i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.38 - 1.95i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.223 - 0.387i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.00 - 6.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.861 + 1.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.570 + 0.987i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.04 - 4.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + (2.59 + 4.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.37 + 2.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.587 + 0.339i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.71iT - 47T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + (-4.67 + 8.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.84 - 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.35 + 9.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.05 - 5.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.118iT - 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (-7.70 + 4.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.84 - 4.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.06392982184148163950531488731, −10.60450468880456106804538680651, −9.021851038859528271989053435685, −8.527325990256431158173273499073, −7.72223585715391145386775750754, −6.42015801392699476507963958212, −5.54067718806699804390222925432, −5.04988890038525012336811849780, −3.83180120909163815780607157285, −2.03841056984391073905833425911, 0.886018413335328616178113363619, 2.13706024994443305806635067001, 3.69686841973136872811935162061, 4.68601647704830538481444674444, 5.64016587718895696627965118191, 6.64633131884868080016245209106, 7.916168206851739271119249442882, 9.053263850280276473761200449926, 9.661566666385859381020768194529, 10.94671515376808560644501072658

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