Properties

Label 2-520-5.4-c1-0-1
Degree $2$
Conductor $520$
Sign $-0.642 - 0.766i$
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  + 1.16i·3-s + (−1.43 − 1.71i)5-s + 4.64i·7-s + 1.64·9-s − 3.25·11-s + i·13-s + (1.99 − 1.66i)15-s − 0.871i·17-s − 6.93·19-s − 5.40·21-s + 6.03i·23-s + (−0.876 + 4.92i)25-s + 5.40i·27-s − 3.63·29-s + 7.78·31-s + ⋯
L(s)  = 1  + 0.671i·3-s + (−0.642 − 0.766i)5-s + 1.75i·7-s + 0.549·9-s − 0.981·11-s + 0.277i·13-s + (0.514 − 0.430i)15-s − 0.211i·17-s − 1.58·19-s − 1.17·21-s + 1.25i·23-s + (−0.175 + 0.984i)25-s + 1.03i·27-s − 0.675·29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384078 + 0.822763i\)
\(L(\frac12)\) \(\approx\) \(0.384078 + 0.822763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 1.71i)T \)
13 \( 1 - iT \)
good3 \( 1 - 1.16iT - 3T^{2} \)
7 \( 1 - 4.64iT - 7T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
17 \( 1 + 0.871iT - 17T^{2} \)
19 \( 1 + 6.93T + 19T^{2} \)
23 \( 1 - 6.03iT - 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 - 7.78T + 31T^{2} \)
37 \( 1 - 8.39iT - 37T^{2} \)
41 \( 1 + 6.44T + 41T^{2} \)
43 \( 1 - 6.01iT - 43T^{2} \)
47 \( 1 + 8.64iT - 47T^{2} \)
53 \( 1 + 4.53iT - 53T^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 + 9.49iT - 67T^{2} \)
71 \( 1 - 8.59T + 71T^{2} \)
73 \( 1 + 3.31iT - 73T^{2} \)
79 \( 1 - 5.18T + 79T^{2} \)
83 \( 1 - 1.66iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 9.62iT - 97T^{2} \)
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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

−11.33980470019829197354361692709, −10.16178414077036500707207287966, −9.377915608127289492362178878186, −8.566700867007289056229642684898, −7.938839789898340814679622716820, −6.50624069449432128117895220298, −5.26767720548617755015328510321, −4.76236788586749221299481027412, −3.47672660338380071436395554341, −2.05135698512435975510062182462, 0.52188902491342571500790953199, 2.35106010345658752839802760093, 3.82193953122671489114638751039, 4.54770068308410282617471722815, 6.34032607081117707096110365541, 7.01727985287180688275100437707, 7.69021522995650702873766343939, 8.369504690294021674047155849566, 10.22615945863263022479840594922, 10.42657187045495668578561665309

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