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Properties

Label 2-520-520.179-c0-0-1
Degree $2$
Conductor $520$
Sign $0.488 + 0.872i$
Analytic cond. $0.259513$
Root an. cond. $0.509424$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + 0.999i·18-s + (1.5 + 0.866i)19-s + (−0.866 − 0.499i)20-s + (−0.866 + 1.5i)22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + 0.999i·18-s + (1.5 + 0.866i)19-s + (−0.866 − 0.499i)20-s + (−0.866 + 1.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(0.259513\)
Root analytic conductor: \(0.509424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :0),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.370939066\)
\(L(\frac12)\) \(\approx\) \(1.370939066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + iT \)
13 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−11.11685484085490249776099682915, −10.18394989924637505603670297279, −9.431227254710861867796914803431, −7.949292908285069192414769351850, −7.64954221051409860725894179835, −5.66888269469929793303333340912, −5.17091462366084332699072593667, −4.60758431602180772088370212134, −2.83380549989469622830659486677, −1.82687579518099703039855486180, 2.61125454976315212669504297765, 3.37523091599560686486990006246, 4.76496495638079557285030163029, 5.61135949594659306578496357635, 6.66822686630699097266992686412, 7.52463476425749293681550711122, 8.123876797951282305719065846222, 9.474221832181350696417102645912, 10.70193344409902959319934820436, 11.36937993591110447375960143854

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