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Properties

Label 1-40e2-1600.909-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.221 - 0.975i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (−0.233 − 0.972i)3-s + (0.707 − 0.707i)7-s + (−0.891 + 0.453i)9-s + (0.760 − 0.649i)11-s + (−0.649 + 0.760i)13-s + (−0.587 + 0.809i)17-s + (0.522 + 0.852i)19-s + (−0.852 − 0.522i)21-s + (0.453 − 0.891i)23-s + (0.649 + 0.760i)27-s + (−0.233 − 0.972i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.0784 − 0.996i)37-s + (0.891 + 0.453i)39-s + ⋯
L(s)  = 1  + (−0.233 − 0.972i)3-s + (0.707 − 0.707i)7-s + (−0.891 + 0.453i)9-s + (0.760 − 0.649i)11-s + (−0.649 + 0.760i)13-s + (−0.587 + 0.809i)17-s + (0.522 + 0.852i)19-s + (−0.852 − 0.522i)21-s + (0.453 − 0.891i)23-s + (0.649 + 0.760i)27-s + (−0.233 − 0.972i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.0784 − 0.996i)37-s + (0.891 + 0.453i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.221 - 0.975i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.221 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9159371634 - 1.147888202i\)
\(L(\frac12)\) \(\approx\) \(0.9159371634 - 1.147888202i\)
\(L(1)\) \(\approx\) \(0.9649176447 - 0.4482916202i\)
\(L(1)\) \(\approx\) \(0.9649176447 - 0.4482916202i\)

Euler product

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

−20.5781980180601785192403077294, −20.181526717993722100048659464032, −19.38262203077079149304757768821, −18.17779073586215369795272543821, −17.60555782588199222329976438799, −17.11572231078764706823353601905, −16.06645522262301365104531712217, −15.3333861028610765331644745502, −14.94185527592461103018707594470, −14.172082329912224003193784149842, −13.16137391038995104144793154358, −12.07124779459366872945034619943, −11.565754424984028102852170286332, −10.93099082379567440969373510616, −9.81752283184567806634663678327, −9.35941516267405006051576350173, −8.60891528326298770342443843751, −7.59155431641265316233901929691, −6.685202434822451638343818849547, −5.58172218449765058951431420312, −4.94659033107989663222536014411, −4.39037771683510726903082567049, −3.14708831800962482758686857038, −2.45937775069209314797127484084, −1.09073127698155660845687550710, 0.64697581875423125165988237700, 1.5991628793748415399667421650, 2.35754350326761190420476703837, 3.68182394365969512117777184476, 4.488469705319819970547923936606, 5.51744633866747531234948041934, 6.431423551414704709714348086791, 6.997943304459852561743254041946, 7.91619085105762390378062030438, 8.49879765002455868226692995547, 9.465078845868977692637739579390, 10.638408751851824955344864002587, 11.20516046053038317377077089685, 11.97906977162717149284396458274, 12.61629969381663355763696786725, 13.60944934578254391885533736310, 14.22856293405844415482369305058, 14.60934798975260917383690479424, 15.95400538698311611300514957393, 16.89771388694326692520163355488, 17.214946662409013699106728299673, 17.92697014242580306468392261195, 19.04455421716335424187508334589, 19.21761640522899645361618006540, 20.17626596813060502545241230267

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