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Properties

Label 1-5200-5200.4221-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.771 + 0.636i$
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  + (0.743 + 0.669i)3-s + (0.5 − 0.866i)7-s + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (0.669 + 0.743i)17-s + (−0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (0.104 − 0.994i)23-s + (−0.587 + 0.809i)27-s + (0.743 + 0.669i)29-s + (0.309 − 0.951i)31-s + (0.669 + 0.743i)33-s + (0.406 − 0.913i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)3-s + (0.5 − 0.866i)7-s + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (0.669 + 0.743i)17-s + (−0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (0.104 − 0.994i)23-s + (−0.587 + 0.809i)27-s + (0.743 + 0.669i)29-s + (0.309 − 0.951i)31-s + (0.669 + 0.743i)33-s + (0.406 − 0.913i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.771 + 0.636i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (4221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.771 + 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.807878319 + 1.009033997i\)
\(L(\frac12)\) \(\approx\) \(2.807878319 + 1.009033997i\)
\(L(1)\) \(\approx\) \(1.572405228 + 0.3154187421i\)
\(L(1)\) \(\approx\) \(1.572405228 + 0.3154187421i\)

Euler product

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

−17.88287331417148999436318508222, −17.46892599123468653574837337669, −16.68644685301490817977917909979, −15.6521982591420221597201553271, −15.16978706332515189725686703830, −14.59523774116139794511221883275, −13.82625197441562755512861769204, −13.487286363721240887054421036683, −12.42849200646954338099770335469, −11.93910138999307392581386210273, −11.54510008473248462932381612624, −10.48750463501369635749692739996, −9.452055761271081910607058549105, −9.145659510524311361135601413834, −8.345321251121538112649579794673, −7.87651410782924679508920660729, −6.89164824869619561578594248842, −6.47272463039744500656448982172, −5.548832471182812052063424452499, −4.77174821782583772321260673985, −3.84134698685354228165778824366, −3.05015321647379206651448951610, −2.38878222267341093286594118200, −1.5849837498754357905332015919, −0.84745134943538715022655316244, 0.95602999157107144356096827175, 1.76987994754383727993937013001, 2.59810636644477482958537597737, 3.62941985824934090379388609580, 4.09932651094794495186067752441, 4.59674080086213624918666879846, 5.60538760235173138848054208072, 6.46707533198266049962098250043, 7.246333170149128028756848774577, 8.0845982614740309702212957333, 8.44361175766801932684137243087, 9.329855880776693839757198645562, 9.95904861787240058729635234737, 10.68724089646037841154658547218, 11.01734837487962532433487172067, 12.15430238847845484261698774417, 12.69908334728164032443829056095, 13.6774806599325735097782107025, 14.19099666152391131139788812954, 14.71280013928857919019105878146, 15.14023529052442282508453719235, 16.19655567567152833040633709821, 16.762660639336097301765075238641, 17.105622370563688888335170318418, 18.01622139606104065454937250777

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