↑

Properties

Label 1-2669-2669.583-r0-0-0
Degree $1$
Conductor $2669$
Sign $0.678 + 0.734i$
Analytic cond. $12.3947$
Root an. cond. $12.3947$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.998 + 0.0603i)2-s + (0.963 + 0.268i)3-s + (0.992 + 0.120i)4-s + (0.728 + 0.685i)5-s + (0.945 + 0.326i)6-s + (−0.871 + 0.491i)7-s + (0.983 + 0.180i)8-s + (0.855 + 0.517i)9-s + (0.685 + 0.728i)10-s + (0.326 − 0.945i)11-s + (0.923 + 0.382i)12-s − i·13-s + (−0.899 + 0.437i)14-s + (0.517 + 0.855i)15-s + (0.970 + 0.239i)16-s + ⋯
L(s)  = 1  + (0.998 + 0.0603i)2-s + (0.963 + 0.268i)3-s + (0.992 + 0.120i)4-s + (0.728 + 0.685i)5-s + (0.945 + 0.326i)6-s + (−0.871 + 0.491i)7-s + (0.983 + 0.180i)8-s + (0.855 + 0.517i)9-s + (0.685 + 0.728i)10-s + (0.326 − 0.945i)11-s + (0.923 + 0.382i)12-s − i·13-s + (−0.899 + 0.437i)14-s + (0.517 + 0.855i)15-s + (0.970 + 0.239i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2669\)    =    \(17 \cdot 157\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(12.3947\)
Root analytic conductor: \(12.3947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2669} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2669,\ (0:\ ),\ 0.678 + 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.087376351 + 2.226930656i\)
\(L(\frac12)\) \(\approx\) \(5.087376351 + 2.226930656i\)
\(L(1)\) \(\approx\) \(2.866380276 + 0.7362860134i\)
\(L(1)\) \(\approx\) \(2.866380276 + 0.7362860134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
157 \( 1 \)
good2 \( 1 + (0.998 + 0.0603i)T \)
3 \( 1 + (0.963 + 0.268i)T \)
5 \( 1 + (0.728 + 0.685i)T \)
7 \( 1 + (-0.871 + 0.491i)T \)
11 \( 1 + (0.326 - 0.945i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.983 - 0.180i)T \)
23 \( 1 + (0.988 - 0.150i)T \)
29 \( 1 + (0.999 + 0.0302i)T \)
31 \( 1 + (-0.768 + 0.640i)T \)
37 \( 1 + (0.999 + 0.0302i)T \)
41 \( 1 + (0.592 + 0.805i)T \)
43 \( 1 + (-0.616 - 0.787i)T \)
47 \( 1 + (-0.935 - 0.354i)T \)
53 \( 1 + (-0.180 + 0.983i)T \)
59 \( 1 + (0.180 + 0.983i)T \)
61 \( 1 + (-0.542 + 0.839i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (-0.977 + 0.209i)T \)
73 \( 1 + (0.963 + 0.268i)T \)
79 \( 1 + (-0.945 - 0.326i)T \)
83 \( 1 + (-0.297 - 0.954i)T \)
89 \( 1 + (0.464 - 0.885i)T \)
97 \( 1 + (0.728 + 0.685i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.51897911471626896011413714745, −18.86005197405246491268588873141, −17.71292895620494246244247471316, −16.86468328105697846029158157079, −16.34097787158462347752891840234, −15.524071495447976215892028126834, −14.6599702581583410195237727755, −14.24115821635451512124290052411, −13.43489810603863708265886327307, −12.82282075855216288640835436460, −12.625427738218570513664165474239, −11.583349599332643384685933491961, −10.49375175466302697038129644279, −9.67380483095050899990611541635, −9.30869667393819914114227853204, −8.26245082896515729577834962880, −7.285754332872249090600421698274, −6.62632064458233504277806854462, −6.16904757977012879008824569024, −4.83144288604923224948370137467, −4.320101126089029370538865309050, −3.57493648033645047286943577363, −2.56661774753421267184622133657, −1.92877831543878729521107551945, −1.15018554331057250218840408356, 1.34516982128632733843510397894, 2.46323153121559761663983147440, 2.9948810203238424785563166355, 3.37247270869870740816440536295, 4.4475973956879421172088013231, 5.423309618145790492376200982228, 6.15943929781164925259152805150, 6.730432780196870164808926328008, 7.59287917471965131583954613718, 8.58991998694423353796527931015, 9.21109147875053850590263964935, 10.34704803222437698761160442482, 10.54028984090626441939134889076, 11.5722668859429670475576230258, 12.7130834082294584044772817702, 13.12959574757906225837678752382, 13.66386970573629832015310218602, 14.55862512646637487466392381538, 14.92260021863652573822296098094, 15.60266910703769036665897440989, 16.3549215116159000762750753723, 17.028420149330129773457038685224, 18.19276811836392424472819371011, 18.90790132124860285789531100735, 19.62345005600458779807418513

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