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Properties

Label 1-3520-3520.1613-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.975 + 0.219i$
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  + (−0.522 − 0.852i)3-s + (0.987 + 0.156i)7-s + (−0.453 + 0.891i)9-s + (0.996 + 0.0784i)13-s + (−0.309 + 0.951i)17-s + (0.522 + 0.852i)19-s + (−0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.996 − 0.0784i)27-s + (−0.233 − 0.972i)29-s + (−0.309 − 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (0.156 + 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (−0.522 − 0.852i)3-s + (0.987 + 0.156i)7-s + (−0.453 + 0.891i)9-s + (0.996 + 0.0784i)13-s + (−0.309 + 0.951i)17-s + (0.522 + 0.852i)19-s + (−0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.996 − 0.0784i)27-s + (−0.233 − 0.972i)29-s + (−0.309 − 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (0.156 + 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.975 + 0.219i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.975 + 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552172803 + 0.1723699008i\)
\(L(\frac12)\) \(\approx\) \(1.552172803 + 0.1723699008i\)
\(L(1)\) \(\approx\) \(1.029831404 - 0.1087349976i\)
\(L(1)\) \(\approx\) \(1.029831404 - 0.1087349976i\)

Euler product

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

−18.30922252328379105186596807075, −17.999927572380265384797085034418, −17.455685012360972022779433652165, −16.494407142107696336910890640092, −15.9803528617323085853078293580, −15.47378836536821749294941512892, −14.50863662315016514038915200476, −14.10161750369951323200576553822, −13.25433999525097130610986383850, −12.2132166090658807371299342318, −11.62980032693715014345161026067, −10.91044883533944695017003176739, −10.59610441240275747989480192266, −9.593705076869801458342655320080, −8.85098374597472435080808204973, −8.3555117925575911275136080483, −7.23052568328863724194361602257, −6.602071618382085738657117249501, −5.557940355178685999439632620242, −5.06700384574776781992130100224, −4.36255880671679476016567132156, −3.58666405300995963926670861431, −2.72075368009045197774316131366, −1.54939300532861869825741970129, −0.596532321507357257079646811141, 0.93673367189013812420331113350, 1.73854857809397374964562430253, 2.24557914735878789509340206434, 3.622517198548587345047270352274, 4.28582009455521452657197075980, 5.43924121437233278081430562702, 5.811724005938392634789947288675, 6.55404090754514781283981860489, 7.58216259370565328653047172000, 8.02779765537240278597110267893, 8.63029197090875548992010141537, 9.67642373478761301579844178657, 10.66197204951202094838845865988, 11.16724327186528433839676894440, 11.82702992972942138073435477426, 12.38326490753574294040235254946, 13.36269079845542793836276907045, 13.712427186069617811537797145976, 14.56804773215604397496819140428, 15.27511113309566279897235858826, 16.158876755551297208921174255889, 16.80371572225895341560184142913, 17.65275448997746393826841718458, 17.92584752339752645903488588717, 18.69975730170570553386877546870

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