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Properties

Label 1-5200-5200.4477-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.991 - 0.129i$
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  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (−0.587 + 0.809i)29-s + (0.587 + 0.809i)31-s + (−0.809 + 0.587i)33-s + (−0.951 + 0.309i)37-s + (−0.951 + 0.309i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.587 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (−0.587 + 0.809i)29-s + (0.587 + 0.809i)31-s + (−0.809 + 0.587i)33-s + (−0.951 + 0.309i)37-s + (−0.951 + 0.309i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (4477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.991 - 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606443404 - 0.1046578455i\)
\(L(\frac12)\) \(\approx\) \(1.606443404 - 0.1046578455i\)
\(L(1)\) \(\approx\) \(0.9919506291 - 0.1404642262i\)
\(L(1)\) \(\approx\) \(0.9919506291 - 0.1404642262i\)

Euler product

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

−17.88526008850423928038083018238, −17.34579975389488589078723851091, −16.756186086506541638091179913730, −16.01026726680122426975690821780, −15.37967794320202405005802068824, −14.788192674675852728279075174986, −14.11895586177452741721681834805, −13.426981068762812906266752272603, −12.24720321784717297729399957803, −11.889935997982526348103840633994, −11.50786686841687934418135073489, −10.507787900002181451008376066815, −9.99191157892891395178609864653, −9.42645280713875446021367933816, −8.57330482149786500479762882806, −7.54547154345160736950692025044, −7.25540553338688587426053764909, −6.12857592711659310750292783377, −5.48274165534998444450045219211, −4.94835376093813414815057289113, −4.13382012826316114515342531463, −3.64347248166026682406005722245, −2.35916456464814484886525348710, −1.57490174751469777086725906524, −0.63372228841021598447576087322, 0.84253390421240601988599319632, 1.41204049141237256777607960415, 2.23994305244887088959153663395, 3.30194060002310908692495636848, 4.15650571815849390972290999584, 5.0874453262759046578680660408, 5.514502943254907173396446895118, 6.28031662283018256379480010380, 6.99240303066134779352554299043, 7.79371983365692573685683645237, 8.29987654022900136405987615036, 9.035717021444197219502235669061, 10.1297582262911209749749330813, 10.80403051178353143363851889120, 11.20984562164166254546780218271, 12.149061097719360279832661703286, 12.248960062803447534521097773400, 13.4247217181521015261555540554, 13.91982108688327920867705565626, 14.4539621578359028258955612287, 15.42885747376749807457282097849, 16.111943095615941168825534948375, 16.7933862992512613443921710560, 17.325088095020024545124965407874, 17.95162488448421565552881256392

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