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Properties

Label 1-5200-5200.61-r0-0-0
Degree $1$
Conductor $5200$
Sign $-0.769 + 0.638i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (−0.994 − 0.104i)3-s + (0.5 − 0.866i)7-s + (0.978 + 0.207i)9-s + (0.207 + 0.978i)11-s + (−0.104 − 0.994i)17-s + (0.994 − 0.104i)19-s + (−0.587 + 0.809i)21-s + (0.978 − 0.207i)23-s + (−0.951 − 0.309i)27-s + (−0.994 − 0.104i)29-s + (−0.809 + 0.587i)31-s + (−0.104 − 0.994i)33-s + (−0.743 − 0.669i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)3-s + (0.5 − 0.866i)7-s + (0.978 + 0.207i)9-s + (0.207 + 0.978i)11-s + (−0.104 − 0.994i)17-s + (0.994 − 0.104i)19-s + (−0.587 + 0.809i)21-s + (0.978 − 0.207i)23-s + (−0.951 − 0.309i)27-s + (−0.994 − 0.104i)29-s + (−0.809 + 0.587i)31-s + (−0.104 − 0.994i)33-s + (−0.743 − 0.669i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ -0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07874727340 + 0.2183261368i\)
\(L(\frac12)\) \(\approx\) \(0.07874727340 + 0.2183261368i\)
\(L(1)\) \(\approx\) \(0.7342961314 - 0.04472383451i\)
\(L(1)\) \(\approx\) \(0.7342961314 - 0.04472383451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.994 - 0.104i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.994 - 0.104i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (-0.743 + 0.669i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−17.56455417764773737901952650501, −17.1448703371359228817340368215, −16.451439569443710755398200716956, −15.77834503303889915747297076514, −15.185371527398591886179530187073, −14.54582034344679418085430780908, −13.64239393954477047042166894916, −12.91749115129251749946155802156, −12.278117838489434177991812566917, −11.60630553972529062898561212660, −11.07447480800944949693549125827, −10.60882516576232981584945103619, −9.52730924084011100124866291704, −9.060823908270918781188371968680, −8.1890168665414117091745976213, −7.475050978872855358530420101613, −6.5836373273876670813809199404, −5.89285443651549669494693121551, −5.395539536455947690777991555913, −4.799301067579608125106982051526, −3.74389766447743132668384129738, −3.15710846086227192357529654343, −1.858730349773251891800538572055, −1.33338649848654106873039880608, −0.073851192024705147178853059577, 1.15017546827771042567907661803, 1.596606709205406167801086905489, 2.78820906568469545998000758573, 3.80787551113303459023527748530, 4.58951021921060853831828749840, 5.04527632468370634379286650545, 5.76347504715985865600311779785, 6.832337569851738583492633035060, 7.28456003178684739809501121015, 7.61893390994360988081170956135, 8.94320611441375185788379848028, 9.5697116988459468948413836128, 10.3293083816758329223445209011, 10.87558980237753483580901126126, 11.62765282964210161643542382094, 11.99746232992900940699478807025, 13.02454062263344391525270662628, 13.33432120672326757297292269424, 14.360475483717008011758623851668, 14.83470049394893370322685291453, 15.84873047185222456372066835599, 16.26234388392999814090678156576, 17.10289075079235755299797565379, 17.45038779016410988158394985392, 18.188823502872547700689555722996

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