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Properties

Label 1-3520-3520.163-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.691 - 0.722i$
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  + (−0.760 − 0.649i)3-s + (−0.453 + 0.891i)7-s + (0.156 + 0.987i)9-s + (0.522 + 0.852i)13-s + (−0.809 + 0.587i)17-s + (−0.760 − 0.649i)19-s + (0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.522 − 0.852i)27-s + (−0.0784 + 0.996i)29-s + (−0.809 − 0.587i)31-s + (0.649 + 0.760i)37-s + (0.156 − 0.987i)39-s + (−0.891 + 0.453i)41-s + (−0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (−0.760 − 0.649i)3-s + (−0.453 + 0.891i)7-s + (0.156 + 0.987i)9-s + (0.522 + 0.852i)13-s + (−0.809 + 0.587i)17-s + (−0.760 − 0.649i)19-s + (0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.522 − 0.852i)27-s + (−0.0784 + 0.996i)29-s + (−0.809 − 0.587i)31-s + (0.649 + 0.760i)37-s + (0.156 − 0.987i)39-s + (−0.891 + 0.453i)41-s + (−0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.691 - 0.722i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02855523974 - 0.06681855868i\)
\(L(\frac12)\) \(\approx\) \(0.02855523974 - 0.06681855868i\)
\(L(1)\) \(\approx\) \(0.6333336010 + 0.03173642637i\)
\(L(1)\) \(\approx\) \(0.6333336010 + 0.03173642637i\)

Euler product

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

−18.91814103188091600469939943696, −18.02251781107280906006959167601, −17.689012819896127710812962953914, −16.73222748312755927893928769844, −16.37354492755508151023870816982, −15.70668176800366121012377350826, −14.976542256891343981697672937154, −14.252115859757113371770263858367, −13.27959061937933698610480676287, −12.84015718544679444203254312320, −11.93934977645475632152478209773, −11.20918299030957227177106242775, −10.4838373545135862524056848298, −10.16553566644689072303053025795, −9.31171256529991845021703591992, −8.47452853603386519448648115422, −7.61471093307909978573230323491, −6.662876653481287071229553604348, −6.20760410062533532235881325184, −5.367836257818287810750968518932, −4.489712917055523352179117045409, −3.89179403802288864185644209581, −3.21055371664710636241043421012, −2.01177710446273589327039739926, −0.78065866026821644965494612865, 0.0303668236307496659178600900, 1.554648485637237981346130199427, 2.02105530039455273289735386941, 3.02716947144894937743527381515, 4.11486468082642819047711817937, 4.88447501792315124970919002372, 5.79123432280972655959819183978, 6.33341662210518081160529741467, 6.86169836634360501269952706680, 7.79993304818792757509656178728, 8.68278899808514017589958333453, 9.183123250499519923707880026350, 10.19966982237967619127812992489, 11.00228381652050435108045629482, 11.57145086468665366801609235616, 12.163574685764049113345712237389, 13.06420658726221827791369577377, 13.27522005365094803300871719833, 14.2818448134909855511408181186, 15.20174476740326620606148345446, 15.78026948017900191469976165669, 16.527250426435592980559738226012, 17.09442186567136604422099694825, 17.94457358205591232388008366402, 18.41362971042153386348834239235

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