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Properties

Label 1-3920-3920.109-r0-0-0
Degree $1$
Conductor $3920$
Sign $0.307 + 0.951i$
Analytic cond. $18.2044$
Root an. cond. $18.2044$
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  + (−0.294 − 0.955i)3-s + (−0.826 + 0.563i)9-s + (0.563 − 0.826i)11-s + (0.433 + 0.900i)13-s + (−0.365 + 0.930i)17-s + (−0.866 − 0.5i)19-s + (0.365 + 0.930i)23-s + (0.781 + 0.623i)27-s + (−0.781 + 0.623i)29-s + (−0.5 − 0.866i)31-s + (−0.955 − 0.294i)33-s + (−0.149 − 0.988i)37-s + (0.733 − 0.680i)39-s + (0.222 − 0.974i)41-s + (0.974 − 0.222i)43-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)3-s + (−0.826 + 0.563i)9-s + (0.563 − 0.826i)11-s + (0.433 + 0.900i)13-s + (−0.365 + 0.930i)17-s + (−0.866 − 0.5i)19-s + (0.365 + 0.930i)23-s + (0.781 + 0.623i)27-s + (−0.781 + 0.623i)29-s + (−0.5 − 0.866i)31-s + (−0.955 − 0.294i)33-s + (−0.149 − 0.988i)37-s + (0.733 − 0.680i)39-s + (0.222 − 0.974i)41-s + (0.974 − 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(18.2044\)
Root analytic conductor: \(18.2044\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5809927247 + 0.4228277463i\)
\(L(\frac12)\) \(\approx\) \(0.5809927247 + 0.4228277463i\)
\(L(1)\) \(\approx\) \(0.8279583852 - 0.1523150223i\)
\(L(1)\) \(\approx\) \(0.8279583852 - 0.1523150223i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.294 - 0.955i)T \)
11 \( 1 + (0.563 - 0.826i)T \)
13 \( 1 + (0.433 + 0.900i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.781 + 0.623i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.149 - 0.988i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.149 - 0.988i)T \)
59 \( 1 + (0.680 + 0.733i)T \)
61 \( 1 + (-0.149 - 0.988i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.433 + 0.900i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
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

−18.14737152964842387020038982332, −17.71026415321630296115669195448, −16.88011608943170902839644602515, −16.43158384341145957743037038980, −15.617905490980421430686282647967, −14.96691186975644648302857659347, −14.617769411562242541441644122736, −13.63800300089479699882230771088, −12.82024393957023575876150726973, −12.10954368577176202509893162620, −11.44034588588205937166399015408, −10.62698059458196776746213979895, −10.215116722165339096025332736, −9.346508993289889898418995124108, −8.82644995919725837399825773310, −8.00009179682638893928393735656, −7.03843561017157092996762078815, −6.276280731017272421293649429382, −5.58466166400809819424957624140, −4.67681424782024682481916865084, −4.2594904644469563264851929390, −3.31226624439622143105629726853, −2.5938597694746532752876540954, −1.45859965688552516629049886290, −0.22846316379200662163798696999, 1.03558803249705138161201975908, 1.7870304782918409483075377899, 2.51305190938746870926964346481, 3.68090397154708165978295795647, 4.22303588228908339864291148076, 5.55121045493803839316695441182, 5.89472781543497268318806618361, 6.78836278553585088752596940965, 7.22767061598620018956118139801, 8.23770307466316524350791272527, 8.82844601294652117859638582238, 9.39060420731488420402336017679, 10.75900414238353830259530210590, 11.14015625155398891925704133785, 11.66004596262649383945182395853, 12.71742885885951812180752326832, 12.997696837637423703910245190849, 13.8862021377716464615463759125, 14.34998549785357598740536413322, 15.19490148066816159567662876470, 16.11773146379636742883004978748, 16.78282162479539876851149582033, 17.31473230968656767606464818682, 17.924739937578741784944391094465, 18.83083840129229828189856469094

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