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Properties

Label 1-2681-2681.226-r0-0-0
Degree $1$
Conductor $2681$
Sign $-0.998 + 0.0580i$
Analytic cond. $12.4505$
Root an. cond. $12.4505$
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  + (0.870 − 0.492i)2-s + (−0.893 − 0.449i)3-s + (0.514 − 0.857i)4-s + (−0.999 − 0.0274i)5-s + (−0.998 + 0.0493i)6-s + (0.0246 − 0.999i)8-s + (0.596 + 0.802i)9-s + (−0.883 + 0.468i)10-s + (0.742 − 0.670i)11-s + (−0.844 + 0.535i)12-s + (−0.977 − 0.212i)13-s + (0.880 + 0.473i)15-s + (−0.471 − 0.882i)16-s + (0.996 + 0.0876i)17-s + (0.914 + 0.404i)18-s + (−0.912 − 0.409i)19-s + ⋯
L(s)  = 1  + (0.870 − 0.492i)2-s + (−0.893 − 0.449i)3-s + (0.514 − 0.857i)4-s + (−0.999 − 0.0274i)5-s + (−0.998 + 0.0493i)6-s + (0.0246 − 0.999i)8-s + (0.596 + 0.802i)9-s + (−0.883 + 0.468i)10-s + (0.742 − 0.670i)11-s + (−0.844 + 0.535i)12-s + (−0.977 − 0.212i)13-s + (0.880 + 0.473i)15-s + (−0.471 − 0.882i)16-s + (0.996 + 0.0876i)17-s + (0.914 + 0.404i)18-s + (−0.912 − 0.409i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2681\)    =    \(7 \cdot 383\)
Sign: $-0.998 + 0.0580i$
Analytic conductor: \(12.4505\)
Root analytic conductor: \(12.4505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2681} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2681,\ (0:\ ),\ -0.998 + 0.0580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04080236396 - 1.404776353i\)
\(L(\frac12)\) \(\approx\) \(-0.04080236396 - 1.404776353i\)
\(L(1)\) \(\approx\) \(0.8488408671 - 0.7163485904i\)
\(L(1)\) \(\approx\) \(0.8488408671 - 0.7163485904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
383 \( 1 \)
good2 \( 1 + (0.870 - 0.492i)T \)
3 \( 1 + (-0.893 - 0.449i)T \)
5 \( 1 + (-0.999 - 0.0274i)T \)
11 \( 1 + (0.742 - 0.670i)T \)
13 \( 1 + (-0.977 - 0.212i)T \)
17 \( 1 + (0.996 + 0.0876i)T \)
19 \( 1 + (-0.912 - 0.409i)T \)
23 \( 1 + (-0.0630 - 0.998i)T \)
29 \( 1 + (0.965 - 0.260i)T \)
31 \( 1 + (0.639 + 0.768i)T \)
37 \( 1 + (0.763 + 0.645i)T \)
41 \( 1 + (-0.668 - 0.743i)T \)
43 \( 1 + (0.991 + 0.131i)T \)
47 \( 1 + (0.166 - 0.986i)T \)
53 \( 1 + (0.230 + 0.972i)T \)
59 \( 1 + (0.304 - 0.952i)T \)
61 \( 1 + (0.613 - 0.789i)T \)
67 \( 1 + (-0.0520 - 0.998i)T \)
71 \( 1 + (-0.983 - 0.179i)T \)
73 \( 1 + (0.133 + 0.990i)T \)
79 \( 1 + (-0.898 - 0.439i)T \)
83 \( 1 + (-0.872 + 0.488i)T \)
89 \( 1 + (0.756 - 0.653i)T \)
97 \( 1 + (-0.509 + 0.860i)T \)
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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

−19.66445562409922293312850551006, −19.13691287472033650293132393639, −17.92269282095683912519355943929, −17.27272312478380284193774369818, −16.69133019931469420620241233535, −16.12232616392013134277882546339, −15.38572707614433900707530953683, −14.73364035383742933595394613670, −14.39277630420895979833056809469, −13.02454131431266587657223668749, −12.41091803680229429530055105397, −11.79731573744047561995686217337, −11.52233676437926699994803107766, −10.43648220033345795501367280422, −9.69662235658864869920638130972, −8.66435659292407024062258051121, −7.623726666596584861413165324488, −7.19523887351828598513419217762, −6.359481247349124843356236482100, −5.62428731715842693675656858096, −4.674338030358167359711900912014, −4.2777871598194419347832079968, −3.58810344421048810914889564419, −2.56261686130633031693371667753, −1.20284258866476362660827921070, 0.43751453480330726632265207730, 1.15924557941409708976796711987, 2.37436789390107873592066590984, 3.19661833209340945538293830388, 4.22186844173666759933666892537, 4.705770753805443858614379371817, 5.541066644240439890030464442887, 6.45387609514597374372220237539, 6.90214748049288754204359235028, 7.8318006915689002144058428487, 8.68673929450906842197000849014, 9.98790189373263705791286139152, 10.5631298878803833490697623766, 11.27595429058865192603513798110, 12.01851082887601494040107213075, 12.27326727531822112891936211336, 12.96285744925305033133481657350, 13.97276169153365536087286470352, 14.56046491804178627134660842632, 15.36918706041895449579102778730, 16.08161736996614957886567080242, 16.77742625066721917013287526365, 17.375170848779567673393341820278, 18.6653849520787644874791976047, 18.98681910654212168218942071614

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