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Properties

Label 1-2675-2675.1036-r1-0-0
Degree $1$
Conductor $2675$
Sign $-0.288 - 0.957i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (−0.867 − 0.498i)2-s + (−0.848 + 0.528i)3-s + (0.503 + 0.864i)4-s + (0.999 − 0.0355i)6-s + (−0.482 − 0.875i)7-s + (−0.00592 − 0.999i)8-s + (0.440 − 0.897i)9-s + (0.986 + 0.165i)11-s + (−0.884 − 0.467i)12-s + (0.523 + 0.851i)13-s + (−0.0177 + 0.999i)14-s + (−0.493 + 0.869i)16-s + (0.780 + 0.625i)17-s + (−0.829 + 0.558i)18-s + (−0.988 + 0.153i)19-s + ⋯
L(s)  = 1  + (−0.867 − 0.498i)2-s + (−0.848 + 0.528i)3-s + (0.503 + 0.864i)4-s + (0.999 − 0.0355i)6-s + (−0.482 − 0.875i)7-s + (−0.00592 − 0.999i)8-s + (0.440 − 0.897i)9-s + (0.986 + 0.165i)11-s + (−0.884 − 0.467i)12-s + (0.523 + 0.851i)13-s + (−0.0177 + 0.999i)14-s + (−0.493 + 0.869i)16-s + (0.780 + 0.625i)17-s + (−0.829 + 0.558i)18-s + (−0.988 + 0.153i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (1036, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ -0.288 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2108823093 - 0.2837186603i\)
\(L(\frac12)\) \(\approx\) \(0.2108823093 - 0.2837186603i\)
\(L(1)\) \(\approx\) \(0.5102029371 + 0.01957677234i\)
\(L(1)\) \(\approx\) \(0.5102029371 + 0.01957677234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.867 - 0.498i)T \)
3 \( 1 + (-0.848 + 0.528i)T \)
7 \( 1 + (-0.482 - 0.875i)T \)
11 \( 1 + (0.986 + 0.165i)T \)
13 \( 1 + (0.523 + 0.851i)T \)
17 \( 1 + (0.780 + 0.625i)T \)
19 \( 1 + (-0.988 + 0.153i)T \)
23 \( 1 + (-0.573 + 0.819i)T \)
29 \( 1 + (-0.553 - 0.832i)T \)
31 \( 1 + (-0.966 - 0.257i)T \)
37 \( 1 + (0.959 - 0.280i)T \)
41 \( 1 + (-0.991 - 0.130i)T \)
43 \( 1 + (0.320 + 0.947i)T \)
47 \( 1 + (-0.182 + 0.983i)T \)
53 \( 1 + (-0.408 + 0.912i)T \)
59 \( 1 + (-0.620 - 0.783i)T \)
61 \( 1 + (-0.683 - 0.729i)T \)
67 \( 1 + (0.949 + 0.314i)T \)
71 \( 1 + (-0.240 + 0.970i)T \)
73 \( 1 + (0.252 - 0.967i)T \)
79 \( 1 + (-0.513 + 0.858i)T \)
83 \( 1 + (0.928 + 0.370i)T \)
89 \( 1 + (-0.275 - 0.961i)T \)
97 \( 1 + (-0.397 - 0.917i)T \)
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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

−19.05095453957957373154475584135, −18.385444567627138611395445659270, −18.15858768549490473067506437426, −17.17214526866215737467950253090, −16.54337712615413615035557944171, −16.17956542903924695178130157712, −15.2050577177504544920290624508, −14.643376580357149265520066258596, −13.65638314281435271723714055438, −12.72957273751674037331901341007, −12.09294752168581587241690330170, −11.41493456949500316424300082785, −10.63953645582802182219829803398, −9.99924875392419215605803436064, −9.05468063514352892966045735794, −8.482759178212625381666686417878, −7.63432697534656637558134082961, −6.76782022380772702525663539205, −6.24211268992443882606243907330, −5.6189138876561509982910164087, −4.92892137838946385134997396326, −3.54446260982438640607578475364, −2.395013388096552738979269550938, −1.57202623094013489725633564767, −0.64001669532163246105295367321, 0.13880132957412093014085455245, 1.20192818939082720960666813209, 1.794642242234730129465114189747, 3.34572755041433276401640836406, 3.95405732410107786795011144353, 4.38915772108413216081117829982, 6.01713639358504787828784516931, 6.359527372104639905019705010257, 7.2351692150972258770120470718, 8.01916339378671761460433368491, 9.1554467336197153679045041880, 9.60846735901457249378679527532, 10.20279112815761586225453410955, 11.18136187583377598546655894755, 11.30903064479934443372653092217, 12.388595935496053612890494904215, 12.835361855498366737757660296679, 13.90491836875689697530091337809, 14.82122295745354099025220210186, 15.72239587690425669135023753452, 16.430691081026868657398394378, 17.04773609235349985704474742584, 17.16225763566082828158692414111, 18.21070289640268417318935510936, 18.95008761545582300748097215727

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