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Properties

Label 1-2888-2888.875-r1-0-0
Degree $1$
Conductor $2888$
Sign $-0.134 + 0.990i$
Analytic cond. $310.358$
Root an. cond. $310.358$
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  + (0.546 − 0.837i)3-s + (−0.789 + 0.614i)5-s + (−0.945 − 0.324i)7-s + (−0.401 − 0.915i)9-s + (−0.401 + 0.915i)11-s + (−0.546 + 0.837i)13-s + (0.0825 + 0.996i)15-s + (0.945 + 0.324i)17-s + (−0.789 + 0.614i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.986 − 0.164i)27-s + (−0.945 − 0.324i)29-s + (0.986 + 0.164i)31-s + (0.546 + 0.837i)33-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)3-s + (−0.789 + 0.614i)5-s + (−0.945 − 0.324i)7-s + (−0.401 − 0.915i)9-s + (−0.401 + 0.915i)11-s + (−0.546 + 0.837i)13-s + (0.0825 + 0.996i)15-s + (0.945 + 0.324i)17-s + (−0.789 + 0.614i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.986 − 0.164i)27-s + (−0.945 − 0.324i)29-s + (0.986 + 0.164i)31-s + (0.546 + 0.837i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(310.358\)
Root analytic conductor: \(310.358\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (875, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2888,\ (1:\ ),\ -0.134 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2390799092 + 0.2737176465i\)
\(L(\frac12)\) \(\approx\) \(0.2390799092 + 0.2737176465i\)
\(L(1)\) \(\approx\) \(0.8162509303 - 0.1651144647i\)
\(L(1)\) \(\approx\) \(0.8162509303 - 0.1651144647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.546 - 0.837i)T \)
5 \( 1 + (-0.789 + 0.614i)T \)
7 \( 1 + (-0.945 - 0.324i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (-0.546 + 0.837i)T \)
17 \( 1 + (0.945 + 0.324i)T \)
23 \( 1 + (-0.546 - 0.837i)T \)
29 \( 1 + (-0.945 - 0.324i)T \)
31 \( 1 + (0.986 + 0.164i)T \)
37 \( 1 + (0.401 - 0.915i)T \)
41 \( 1 + (-0.0825 - 0.996i)T \)
43 \( 1 + (-0.677 - 0.735i)T \)
47 \( 1 + (0.401 - 0.915i)T \)
53 \( 1 + (0.401 - 0.915i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (0.879 + 0.475i)T \)
67 \( 1 + (-0.879 + 0.475i)T \)
71 \( 1 + (0.879 - 0.475i)T \)
73 \( 1 + (0.945 + 0.324i)T \)
79 \( 1 + (0.677 + 0.735i)T \)
83 \( 1 + (0.789 + 0.614i)T \)
89 \( 1 + (0.945 - 0.324i)T \)
97 \( 1 + (-0.879 - 0.475i)T \)
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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

−19.107172881265249307146881383902, −18.25863723877029366416812413306, −17.00911541480027056857291670226, −16.478083572687568661076969007467, −16.00952493718158272300384533657, −15.29324991257642298331063976598, −14.875529987348080232462038410357, −13.72227477142885302409515510992, −13.243244725984071921687371933234, −12.38933611333519203451444838598, −11.68437719903948723433073809405, −10.88800601501575758791498459721, −9.96496724941108758914350186123, −9.54124670264797711777652512216, −8.71642811860159442515156479774, −7.932542815879785095279590338931, −7.59719881963925688544674290834, −6.167666891152248715381905579180, −5.42739935041176287235532917984, −4.801398113335455918073782594410, −3.758970451514661185145646664161, −3.19969635438681171196748364108, −2.65077860262910496276556387169, −1.11296039125479967793168093362, −0.07971431140839572517301641934, 0.683213102384132188774656846822, 2.08854168430843374430754975745, 2.53831824202918959913300074513, 3.672468862810005493767799931748, 3.977695296122585946712280475320, 5.287382113986299857951483477, 6.45223274083787274585681152366, 6.86286564673040723323595811891, 7.53384308680866847691846910601, 8.09993705119891708335315178898, 9.08106707144360495815371702634, 9.90320668938231132764950327878, 10.43659126966971147352179137821, 11.596010584713288384898349709794, 12.250023917155541602958144145432, 12.62507394187041949756223031813, 13.56204235210892797609998440599, 14.25297186605229874146119509417, 14.87485337946611787183026283752, 15.48940757318258186018247347404, 16.382980195372893698708261909225, 17.054979128273631607566749080067, 17.97869793565133463951443897356, 18.706247819805300994753364833578, 19.11611029236269136726005520233

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