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Properties

Label 1-40e2-1600.21-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.191 + 0.981i$
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  + (−0.649 − 0.760i)3-s + (0.707 + 0.707i)7-s + (−0.156 + 0.987i)9-s + (−0.522 − 0.852i)11-s + (−0.852 − 0.522i)13-s + (−0.951 − 0.309i)17-s + (−0.996 + 0.0784i)19-s + (0.0784 − 0.996i)21-s + (0.987 − 0.156i)23-s + (0.852 − 0.522i)27-s + (0.649 + 0.760i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.233 − 0.972i)37-s + (0.156 + 0.987i)39-s + ⋯
L(s)  = 1  + (−0.649 − 0.760i)3-s + (0.707 + 0.707i)7-s + (−0.156 + 0.987i)9-s + (−0.522 − 0.852i)11-s + (−0.852 − 0.522i)13-s + (−0.951 − 0.309i)17-s + (−0.996 + 0.0784i)19-s + (0.0784 − 0.996i)21-s + (0.987 − 0.156i)23-s + (0.852 − 0.522i)27-s + (0.649 + 0.760i)29-s + (−0.309 + 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.233 − 0.972i)37-s + (0.156 + 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.191 + 0.981i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.191 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2403595132 + 0.2917090465i\)
\(L(\frac12)\) \(\approx\) \(0.2403595132 + 0.2917090465i\)
\(L(1)\) \(\approx\) \(0.7010931121 - 0.09354330555i\)
\(L(1)\) \(\approx\) \(0.7010931121 - 0.09354330555i\)

Euler product

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

−20.427117931100608121882449033646, −19.68494525513410217262659564848, −18.698515853871862257564983418093, −17.68488439942934257665881715848, −17.26283797718060548124552197481, −16.77178215663225019926495760403, −15.71451718743052654434385635131, −14.98901016856993037461814019954, −14.61263539946043641179636258396, −13.38913976180707610326946816123, −12.71626194159015581098363458062, −11.66318104085327978048969780739, −11.18601236809532319448434539693, −10.317438881317097348927475982304, −9.80912873558719424030527327638, −8.85045269251600305255868065982, −7.902685176069248956356370254409, −6.95424589402257316901172167079, −6.32192713120852356284538787252, −4.99906640689873347095752094661, −4.649768703644963701360739263243, −3.9708665357765485069078144514, −2.63393783797500494000303551462, −1.629981219363580758066518602042, −0.16004020697156262565080395070, 1.116968478671318074519478418088, 2.270693823121830533784014285082, 2.80961574745144945650639825334, 4.38512358917434710194214510690, 5.296060985615018533770123327720, 5.66395099873022346026494577767, 6.84052064990545630833184276922, 7.38043405338425286302393353100, 8.603590416384372141498838940842, 8.7134010256831617154232529124, 10.386760388224945741473380954153, 10.85643944768863198829862471223, 11.61890530853058410827429196130, 12.4436210423159155629481443161, 12.94228108621253105130718576318, 13.866799924362644079768019113311, 14.63670105203316481471603784858, 15.51242084281629677003907912001, 16.25569615776138299840003071849, 17.227085094226638610915582694222, 17.672956802616253875036282880191, 18.43295492680897421061621390714, 19.05149666133827726264599347534, 19.74070125187184671815431996177, 20.72358829574923988538966620570

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