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Properties

Label 1-520-520.259-r1-0-0
Degree $1$
Conductor $520$
Sign $1$
Analytic cond. $55.8817$
Root an. cond. $55.8817$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 17-s − 19-s + 21-s + 23-s − 27-s − 29-s + 31-s + 33-s − 37-s − 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s − 59-s − 61-s − 63-s + 67-s − 69-s + 71-s + 73-s + ⋯
L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 17-s − 19-s + 21-s + 23-s − 27-s − 29-s + 31-s + 33-s − 37-s − 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s − 59-s − 61-s − 63-s + 67-s − 69-s + 71-s + 73-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(55.8817\)
Root analytic conductor: \(55.8817\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{520} (259, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 520,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5148202990\)
\(L(\frac12)\) \(\approx\) \(0.5148202990\)
\(L(1)\) \(\approx\) \(0.5510718060\)
\(L(1)\) \(\approx\) \(0.5510718060\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−23.09707377972868128018937781483, −22.72418238364784555759794173295, −21.70491731815742168912800708006, −21.0482588132289447096488611110, −19.88298558957835339677332766670, −18.907317891868830967075490635568, −18.33069348129050378369843404392, −17.222503621981959000857993914435, −16.67709708897373370975893943168, −15.61706518541201909087792985322, −15.22090808881279009348537786166, −13.45200329371061067476258440370, −13.01908459021767277876184131691, −12.14952871468906681125493356351, −11.04508347773828008508857891623, −10.41596331304031995801954993560, −9.511526498722929217846882569523, −8.36237745467781401456136119648, −7.02788207198529074584542600041, −6.46796012170503251356478368880, −5.409260178176494348875078319853, −4.53173229427681136337963490431, −3.29460287980216002325118225496, −1.99999002129155580385294339465, −0.38464189291356244580902717207, 0.38464189291356244580902717207, 1.99999002129155580385294339465, 3.29460287980216002325118225496, 4.53173229427681136337963490431, 5.409260178176494348875078319853, 6.46796012170503251356478368880, 7.02788207198529074584542600041, 8.36237745467781401456136119648, 9.511526498722929217846882569523, 10.41596331304031995801954993560, 11.04508347773828008508857891623, 12.14952871468906681125493356351, 13.01908459021767277876184131691, 13.45200329371061067476258440370, 15.22090808881279009348537786166, 15.61706518541201909087792985322, 16.67709708897373370975893943168, 17.222503621981959000857993914435, 18.33069348129050378369843404392, 18.907317891868830967075490635568, 19.88298558957835339677332766670, 21.0482588132289447096488611110, 21.70491731815742168912800708006, 22.72418238364784555759794173295, 23.09707377972868128018937781483

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