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Properties

Label 1-40e2-1600.539-r1-0-0
Degree $1$
Conductor $1600$
Sign $-0.975 - 0.221i$
Analytic cond. $171.943$
Root an. cond. $171.943$
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

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

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.975 - 0.221i$
Analytic conductor: \(171.943\)
Root analytic conductor: \(171.943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (1:\ ),\ -0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02455039728 + 0.2184414990i\)
\(L(\frac12)\) \(\approx\) \(0.02455039728 + 0.2184414990i\)
\(L(1)\) \(\approx\) \(0.8047727598 + 0.1821141921i\)
\(L(1)\) \(\approx\) \(0.8047727598 + 0.1821141921i\)

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

−19.87912737629173611280733460399, −18.971040455282296104333226273553, −18.52662797191171327939544804966, −17.77623897588138458355685870561, −16.97003017948908328842699494051, −16.24596141338684082488431489491, −15.44906228266875350665320802188, −14.54422493883554353890350914092, −13.68645643895297791776529411088, −13.059535917920659160077766562974, −12.37117670088827333389610408033, −11.67798639827342192253883825883, −10.939482474604858072094898692266, −9.96550689345802330867123153505, −8.88144433686432785970136547353, −8.408738159230496161301387245, −7.46157939021414887035261430105, −6.50868910485402029444516403441, −5.916509298737656310093762189709, −5.38552338803079928822792945918, −3.75240489565769185188033402717, −3.16822477278416661978127862624, −1.9937282844747164113374018172, −1.18874347063910153903671514262, −0.0493808364298796527114622658, 0.97845067395839500417585569191, 2.41447993885835568323907552166, 3.40022244265204272744405646474, 4.28434899648047484436975682654, 4.64921135959858366712916107446, 5.97631108774980149930692767134, 6.6247316272998388927326551720, 7.39427481841347038409256466185, 8.72729698841577229962339000616, 9.373087506198105855915633472267, 9.859182372396719521621289509961, 10.86356569888471715137802395934, 11.40676591080464869775433221098, 12.24735442458345347832906395737, 13.26737245246645383706815239996, 14.04776910006858941508581091287, 14.66791624996621842392656949264, 15.69204072492100976610327206395, 16.22010066111545809248986085747, 16.735232131215666028426867832085, 17.67571784372086150783412704366, 18.23024814559459152222556127030, 19.70779846337344670049675501460, 19.769774909166682924362949457152, 20.732622671718873233382691051606

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