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Properties

Label 1-5200-5200.909-r0-0-0
Degree $1$
Conductor $5200$
Sign $-0.353 - 0.935i$
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  + (0.587 + 0.809i)3-s − 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)17-s + (−0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 − 0.951i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s − 7-s + (−0.309 + 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)17-s + (−0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (−0.951 + 0.309i)27-s + (−0.587 − 0.809i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.951 − 0.309i)37-s + (0.309 − 0.951i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ -0.353 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1591337082 + 0.2302473090i\)
\(L(\frac12)\) \(\approx\) \(-0.1591337082 + 0.2302473090i\)
\(L(1)\) \(\approx\) \(0.8097662453 + 0.3880229669i\)
\(L(1)\) \(\approx\) \(0.8097662453 + 0.3880229669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.587 - 0.809i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
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

−17.618638239594880791790009035365, −16.845271866422248676595296232156, −16.19065899063372258471965905201, −15.47667982394185029489757363907, −14.85697486326682897332211401552, −14.0639987431940909934570384270, −13.4036605193247289563566189497, −12.95352204565013144540133651914, −12.39266302421154932522116157548, −11.650692040229583200521155034151, −10.744357973814524920191999691805, −10.03342573440098493277334852375, −9.32705674974256647354670215217, −8.624364772156022790706233779814, −8.040481093039574470743345297099, −7.16299500754870618160823175050, −6.745860432690136816240715259173, −5.94535306174806910304925915295, −5.2002401611216680016233643917, −4.19556418203684784291191520339, −3.096668459116246617503413818189, −2.92326672257909275093852469080, −2.03434251671589897395655642657, −0.913315070139918145199579450751, −0.07311665739083270829529705073, 1.52775303698000006889839704441, 2.41089075459417049338193741408, 3.19319208842005128744818491916, 3.68650145129526291549547052444, 4.49123747500878656456938085110, 5.39821808319238567959356169679, 5.90382772341179875895915445156, 6.884821591756152477293218947482, 7.81135797311573842323547803760, 8.18028969993051112890995398329, 9.1681660405845793983305071201, 9.698551014958556183458502345615, 10.32828144395725257400022324052, 10.71917812682019683777330694981, 11.78325533862264886720372992090, 12.59001842620324924183678976650, 13.159943542927571382986158179230, 13.77918022493545222295261547841, 14.61616430750258924763788331134, 15.140389546460496899850234690197, 15.89178293134869906160369567057, 16.18237505921210119961489388513, 17.03516475230453542811432092575, 17.61117296155816326432389451196, 18.705866091922541815445557883248

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