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Properties

Label 1-2600-2600.2203-r1-0-0
Degree $1$
Conductor $2600$
Sign $0.994 - 0.101i$
Analytic cond. $279.408$
Root an. cond. $279.408$
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 + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (−0.994 − 0.104i)23-s + (−0.587 + 0.809i)27-s + (0.669 − 0.743i)29-s + (0.951 + 0.309i)31-s + (−0.669 − 0.743i)33-s + (−0.913 − 0.406i)37-s + (0.406 − 0.913i)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 + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)19-s + (0.951 − 0.309i)21-s + (−0.994 − 0.104i)23-s + (−0.587 + 0.809i)27-s + (0.669 − 0.743i)29-s + (0.951 + 0.309i)31-s + (−0.669 − 0.743i)33-s + (−0.913 − 0.406i)37-s + (0.406 − 0.913i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.994 - 0.101i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (2203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ 0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.339223484 - 0.1187538316i\)
\(L(\frac12)\) \(\approx\) \(2.339223484 - 0.1187538316i\)
\(L(1)\) \(\approx\) \(1.243930378 + 0.1718044838i\)
\(L(1)\) \(\approx\) \(1.243930378 + 0.1718044838i\)

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 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (0.406 - 0.913i)T \)
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 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (0.207 - 0.978i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.994 + 0.104i)T \)
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

−19.0883636984708645265342051313, −18.55073866155208616425214444214, −17.82456512120739067590568194934, −17.519722875446178202354173085375, −16.11691255625972708142513327428, −15.48077485414154517074640711953, −15.0508523815550631540643448196, −14.04434027105298839008797878205, −13.62799243627934237876009095645, −12.710989724647581308011450700741, −12.2270994142165876522623055014, −11.39781278968045066627489005570, −10.573370587379133890055275646011, −9.57675517202422777324263712148, −8.881055341413516477705622489932, −8.23753891589927645292879000574, −7.66799552365207091331584522352, −6.72763886145988616342824695618, −6.049097406485884948212620268808, −5.02194585823487791511929924108, −4.33461856892862238558260862539, −3.041338071375949931163688652960, −2.4803582529539040834987388474, −1.84639314592465171168354166501, −0.66043714409342871696510871064, 0.44794031927940304228083489502, 1.841720419943857144837507753984, 2.415858352835329359917243240628, 3.550155700071560297619895607631, 4.18945941134574575380174560661, 4.79898519906282082831507463254, 5.77014105703560941988832533070, 6.75441162060925138994827930161, 7.82154760408756178604666110561, 8.14081998902086492883129045137, 8.88798430628828941244166468424, 9.968279550998173332583015272491, 10.50698060420513722363715674635, 10.86156115668959012759122115546, 12.024394912403574534355053847701, 12.92324920000274813921187848634, 13.733902854365683443553107493789, 14.06450467236826423629668731514, 14.98705478209417092059938154849, 15.58403887296475908730795380999, 16.21301450918806852168421961049, 17.04705234461809436703023476300, 17.67309501490499563816427120427, 18.54035738906317944035059249729, 19.44182774156413462984606362534

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