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Properties

Label 1-2800-2800.221-r0-0-0
Degree $1$
Conductor $2800$
Sign $-0.888 - 0.458i$
Analytic cond. $13.0031$
Root an. cond. $13.0031$
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 + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (−0.587 − 0.809i)13-s + (−0.978 + 0.207i)17-s + (−0.743 + 0.669i)19-s + (−0.913 + 0.406i)23-s + (−0.587 + 0.809i)27-s + (−0.951 + 0.309i)29-s + (−0.978 + 0.207i)31-s + (0.669 + 0.743i)33-s + (−0.994 + 0.104i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)3-s + (0.104 + 0.994i)9-s + (0.994 + 0.104i)11-s + (−0.587 − 0.809i)13-s + (−0.978 + 0.207i)17-s + (−0.743 + 0.669i)19-s + (−0.913 + 0.406i)23-s + (−0.587 + 0.809i)27-s + (−0.951 + 0.309i)29-s + (−0.978 + 0.207i)31-s + (0.669 + 0.743i)33-s + (−0.994 + 0.104i)37-s + (0.104 − 0.994i)39-s + (0.809 − 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(13.0031\)
Root analytic conductor: \(13.0031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2800,\ (0:\ ),\ -0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07041698804 + 0.2903054462i\)
\(L(\frac12)\) \(\approx\) \(-0.07041698804 + 0.2903054462i\)
\(L(1)\) \(\approx\) \(0.9821077188 + 0.2886138940i\)
\(L(1)\) \(\approx\) \(0.9821077188 + 0.2886138940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.743 + 0.669i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.951 + 0.309i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.994 + 0.104i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−19.00342704983371438527803814154, −18.12255239937160696161122223301, −17.51713395331579222802694220173, −16.800438464986213688449020785293, −15.95374233735872177661026897421, −15.071435291140330914090899306870, −14.43121791663826119912098361842, −14.01669829528547867462054006958, −13.040369315430169308771797554561, −12.63508580331387420945935954415, −11.59828684421043346955390524916, −11.24988879019840573862179214771, −9.951594305684852802586535526414, −9.22930502078723143932467553068, −8.80280652771760843924064858878, −7.93254892404330019703126702431, −7.02482241268894423184893243343, −6.63412185974758961502053552894, −5.79091311266636656197023063907, −4.41230175988155383833333431173, −4.041513716937966374207945282959, −2.90563249568895285795414069924, −2.107457914242849354201072903412, −1.48409088678337600398748072046, −0.07189573852227076854055706721, 1.73952271319116382339329933181, 2.2055335930048753266319864789, 3.50852738257766218441315452705, 3.81405449270372673471029928251, 4.78270709334687740043021706164, 5.55694470684670017105248537201, 6.520326543761400657699949920671, 7.42159074816586385957360368834, 8.12978205270017833026864208768, 8.93168867166830925004103431219, 9.43433367471107274359133001483, 10.318012486164295169002293627727, 10.81691611194222280028418008498, 11.772558803531445264716492569329, 12.62472776452460645629541407903, 13.28455188649812706242120696351, 14.19943140964314301813944325042, 14.67743673443185041382703598399, 15.28813756860618818551192109234, 16.0098126632033438549693920324, 16.773253842303658127548548796166, 17.418183529607312815974077061087, 18.179135348952390407613920785, 19.28491158563860510288119545414, 19.57798148086939958993655863346

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