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Properties

Label 1-60e2-3600.3397-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.340 - 0.940i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (0.866 + 0.5i)7-s + (0.406 + 0.913i)11-s + (−0.913 − 0.406i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (−0.994 − 0.104i)23-s + (0.207 − 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (−0.5 + 0.866i)43-s + (0.207 − 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.406 − 0.913i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.406 + 0.913i)11-s + (−0.913 − 0.406i)13-s + (0.951 + 0.309i)17-s + (0.951 + 0.309i)19-s + (−0.994 − 0.104i)23-s + (0.207 − 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (−0.5 + 0.866i)43-s + (0.207 − 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.406 − 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.340 - 0.940i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494566134 - 1.048451381i\)
\(L(\frac12)\) \(\approx\) \(1.494566134 - 1.048451381i\)
\(L(1)\) \(\approx\) \(1.129030903 + 0.02219911336i\)
\(L(1)\) \(\approx\) \(1.129030903 + 0.02219911336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (0.207 - 0.978i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (-0.406 - 0.913i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−18.475126320951836338202072794138, −18.10143568886333952024301184959, −17.188760158707081923849238183184, −16.56810558715333457735747273778, −16.179466369777879430726059891, −15.01651756337801993066965868143, −14.45306951830600117124052754517, −13.93760152309770454170802039239, −13.32577918492885679589950213412, −12.16936894577608204098247719905, −11.74981193792669064159710585636, −11.10455838213195945014533182867, −10.24181342241155270594511785608, −9.61009264067034502522343628667, −8.7815236805787378352699971237, −7.99609490398146321785080599350, −7.38924277425804304050602119019, −6.702451064329261571377375407615, −5.593480332428211680651013579695, −5.131280388230193419884201463786, −4.19826684582309747519531031053, −3.4608783713841705702503939434, −2.58416994440543477657329928488, −1.50408086308773596889804595339, −0.88253431512706783344856305228, 0.29761238265098211976429213669, 1.52204097589978435944096672048, 2.05792014196261611581587073859, 3.037083766421098453091342256924, 3.984628214582301179637210330223, 4.78855003945440069152046565146, 5.443749649644792471609313656759, 6.12936611690503763041219131411, 7.281545280430044570678296827, 7.72786903064593629869509385479, 8.39356898484833868641853311323, 9.46072002280328781320686837613, 9.86427371047623653291455507830, 10.66589327457241529494399034950, 11.671086251559191915755906665672, 12.11201854003392454568854896958, 12.60203926972651165954463547325, 13.70949309283079648455727739119, 14.39959104359567695456288051445, 14.90306006504157243442635649630, 15.45863173101554357865486325714, 16.45074352607982348899022320891, 17.04349977701812153157128710206, 17.83756309122855384561854915317, 18.20846567803684713330679337531

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