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Properties

Label 1-3520-3520.1333-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.582 - 0.812i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (0.996 − 0.0784i)3-s + (−0.891 − 0.453i)7-s + (0.987 − 0.156i)9-s + (0.972 + 0.233i)13-s + (−0.809 + 0.587i)17-s + (0.996 − 0.0784i)19-s + (−0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (0.972 − 0.233i)27-s + (−0.649 − 0.760i)29-s + (0.809 + 0.587i)31-s + (0.0784 − 0.996i)37-s + (0.987 + 0.156i)39-s + (−0.453 − 0.891i)41-s + (0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (0.996 − 0.0784i)3-s + (−0.891 − 0.453i)7-s + (0.987 − 0.156i)9-s + (0.972 + 0.233i)13-s + (−0.809 + 0.587i)17-s + (0.996 − 0.0784i)19-s + (−0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (0.972 − 0.233i)27-s + (−0.649 − 0.760i)29-s + (0.809 + 0.587i)31-s + (0.0784 − 0.996i)37-s + (0.987 + 0.156i)39-s + (−0.453 − 0.891i)41-s + (0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.582 - 0.812i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.582 - 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.059767670 - 1.057490034i\)
\(L(\frac12)\) \(\approx\) \(2.059767670 - 1.057490034i\)
\(L(1)\) \(\approx\) \(1.412070965 - 0.2211953230i\)
\(L(1)\) \(\approx\) \(1.412070965 - 0.2211953230i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.996 - 0.0784i)T \)
7 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (0.972 + 0.233i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.996 - 0.0784i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.649 - 0.760i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.0784 - 0.996i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.852 + 0.522i)T \)
59 \( 1 + (0.996 + 0.0784i)T \)
61 \( 1 + (0.522 - 0.852i)T \)
67 \( 1 + (-0.382 - 0.923i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + (0.453 - 0.891i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + (-0.522 + 0.852i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−18.83165903219383915536524254437, −18.31659378454345004034616072618, −17.71263184751742734174109613730, −16.42595437120702701921181505159, −15.98833576520468661409457987990, −15.4796422626321831155275882412, −14.76415559327233836350964630011, −13.89824577690329645113549312911, −13.24830738209273736931939004778, −12.99510785877979870616201492854, −11.827009988289539990717535310995, −11.30260475403818381555798388519, −10.04091777043104946056620617363, −9.7889862532925679004108957196, −8.9335384753648988414516279490, −8.40536747554816573522116524335, −7.56978644420469495397306653053, −6.80067620481774954708858128760, −6.07316377831278488215093850220, −5.17177172264215149746746607851, −4.20007115484689680647169943434, −3.378560384884981719839349702626, −2.93002051275435008802084090236, −1.99283394856616898841390291885, −1.04155632606951942936335165228, 0.64119685132187678103831081598, 1.72846932783638985644635546780, 2.50338916039663902405082899542, 3.4866407109448013563146978682, 3.86311172521176606206733065018, 4.71887119605563614186693590595, 6.01936407497787849601164018744, 6.52833484406972201578823019349, 7.3639471703504291869511430926, 8.02742261891655122490287788520, 8.90177427809880919386040627490, 9.31476760608815987003813420146, 10.22023420591326976788678587566, 10.717431286023763237023612392132, 11.76445421956608662763406503730, 12.64194481404699215669201660718, 13.17314628655162600013333109574, 13.89142230945283419853709011688, 14.216732891032743819978955578998, 15.381534357892977139380996955679, 15.77697659925912919906897596510, 16.33916594675238672852509818950, 17.31543071484366607091103322132, 18.086362800585041022525490032378, 18.77721560725233955446224241260

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