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Properties

Label 1-520-520.123-r1-0-0
Degree $1$
Conductor $520$
Sign $-0.558 - 0.829i$
Analytic cond. $55.8817$
Root an. cond. $55.8817$
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)3-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)33-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(55.8817\)
Root analytic conductor: \(55.8817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 520,\ (1:\ ),\ -0.558 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304245432 - 2.449312021i\)
\(L(\frac12)\) \(\approx\) \(1.304245432 - 2.449312021i\)
\(L(1)\) \(\approx\) \(1.361780025 - 0.6730081530i\)
\(L(1)\) \(\approx\) \(1.361780025 - 0.6730081530i\)

Euler product

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

−23.77359292232714143391574479333, −22.47608031297158998412574581503, −21.87104390891581776823788836199, −21.07952377120527745024516442490, −20.32422612919653750547107695360, −19.424783340916188700753798740012, −18.7802324071622867830215083686, −17.66965139807428601327834298745, −16.81221738102755126131191424012, −15.70825616872633954307717677873, −14.87896287422200218061234238798, −14.592613054572653508695733349680, −13.33337002454861408243848402183, −12.515685720272792376105825096335, −11.394575449690213985595018088110, −10.53739678565581362571974781392, −9.31372870782373480492577463355, −8.883107571663334770655310029838, −7.934657487909729686197701748027, −6.84443320574512924177904016608, −5.60539430132170016942202624114, −4.51183020652589788230474644098, −3.70924313875084204398289701920, −2.392133693151336590880811926859, −1.64421771867151098377724345601, 0.59835288864972498728627405, 1.63238358796977166125121417296, 2.81410085901913442771566811312, 3.91220440458630257758152297284, 4.75849754803225631675786883651, 6.48938782912598550307571335914, 6.97754550021958572288958925842, 8.14148170779946202285743992436, 8.803002782075202732023899984649, 9.76637068351208417972823309719, 10.961028110345439342770049776818, 11.72194347269559904361607803452, 13.03066873700883512931547161340, 13.52785688997862777284323689226, 14.47543608352016542804832212444, 15.029139534410500014680203874241, 16.26616878000587117145891654658, 17.2517748943842509414996201549, 17.93240619179196208250668603963, 19.06601943211990563575831279823, 19.62197123437172493098742312002, 20.45815929694562080861775379843, 21.12618566966737446414625215495, 22.16846225197706753199543532114, 23.23200664066446945281065867306

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