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Properties

Label 1-3520-3520.1347-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.843 + 0.536i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.843 + 0.536i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.843 + 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.394936340 + 0.6973946145i\)
\(L(\frac12)\) \(\approx\) \(2.394936340 + 0.6973946145i\)
\(L(1)\) \(\approx\) \(1.472508731 + 0.1684759118i\)
\(L(1)\) \(\approx\) \(1.472508731 + 0.1684759118i\)

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

−18.76119030993191594237357997715, −18.31649391390046524287475033291, −17.17096173662079722704047753515, −16.50697602743070593773014344440, −15.93816438774277774477341887730, −15.10512093227925511061059939723, −14.623809516843882548146442352031, −13.63036610256823841935925028514, −13.28624954589115626441437118526, −12.75247091257204739444174235654, −11.72229770099812083583729387455, −10.943133029509675148483666650355, −10.01432401041776509347389427366, −9.57226690359941810309080650420, −8.84653246303071529089753396388, −8.08264167175725547133074031535, −7.37976097852796696283699446468, −6.66231699253172305571066855031, −5.972987574039145091291829108911, −4.86591968966381842651951534627, −3.83196962993201030094648809116, −3.520026455561396396069709557592, −2.63138925082432439764167457741, −1.710379084795445778122083447777, −0.760358960953506893456114195093, 0.94637851745602967884335335796, 2.00330415245934618374329904678, 2.710562958713216428973936607106, 3.65402810898521288517949860052, 3.90387036696546261440309434452, 5.21365467290781761246287640716, 5.95321663842865370431724151954, 6.81065915850532220856669684020, 7.44833913264379084868337718644, 8.512918201272231921719971566273, 8.82081424336922073157873376028, 9.48621846615604652304116131438, 10.56710912492792629196068454092, 10.70100516439283717241139538875, 12.14155283673633578733537327430, 12.772721028294265227320525516790, 13.12210804096274031152602732963, 14.0218549446958175421804177660, 14.78183682208093747188453187037, 15.1905939365112210504833836786, 16.12852734083826286485108418888, 16.41339468477656850125244727691, 17.46565221707024092001035926248, 18.4311387502286186706133445667, 18.87228998758022243890395336570

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