Properties

Label 2-520-5.4-c1-0-9
Degree $2$
Conductor $520$
Sign $0.966 + 0.256i$
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  − 1.76i·3-s + (2.16 + 0.573i)5-s + 2.87i·7-s − 0.128·9-s + 4.07·11-s + i·13-s + (1.01 − 3.82i)15-s + 6.32i·17-s − 5.46·19-s + 5.07·21-s − 4.09i·23-s + (4.34 + 2.47i)25-s − 5.07i·27-s + 8.31·29-s − 2.82·31-s + ⋯
L(s)  = 1  − 1.02i·3-s + (0.966 + 0.256i)5-s + 1.08i·7-s − 0.0428·9-s + 1.22·11-s + 0.277i·13-s + (0.261 − 0.987i)15-s + 1.53i·17-s − 1.25·19-s + 1.10·21-s − 0.853i·23-s + (0.868 + 0.495i)25-s − 0.977i·27-s + 1.54·29-s − 0.508·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.966 + 0.256i$
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: \((2,\ 520,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78432 - 0.232696i\)
\(L(\frac12)\) \(\approx\) \(1.78432 - 0.232696i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 0.573i)T \)
13 \( 1 - iT \)
good3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 - 2.87iT - 7T^{2} \)
11 \( 1 - 4.07T + 11T^{2} \)
17 \( 1 - 6.32iT - 17T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 + 4.09iT - 23T^{2} \)
29 \( 1 - 8.31T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 7.77iT - 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 6.06iT - 43T^{2} \)
47 \( 1 + 6.87iT - 47T^{2} \)
53 \( 1 + 1.24iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 8.22T + 61T^{2} \)
67 \( 1 - 5.38iT - 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 2.77iT - 73T^{2} \)
79 \( 1 + 9.83T + 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 0.205iT - 97T^{2} \)
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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

−10.79473308948554355091906311085, −9.973039147832706124896688839856, −8.804480344359136096994777619309, −8.413235878388201945261824302085, −6.70567227539753645152297889154, −6.54811424995111377653004150812, −5.55480048449321506828593782923, −4.04591301687035372064807842114, −2.34229342734450385397590730405, −1.62422947273143985885898424055, 1.33863381248661066944391940406, 3.17890192307839114023878828722, 4.36485765676181110993591778180, 4.96929421142375885460975700137, 6.35366651968674511040513898787, 7.09493525806972053407157591388, 8.509321553406316883838025620868, 9.455986191618001525742870805445, 9.925008551259458198396248423877, 10.66062145036807345969301833236

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