↑

Properties

Label 1-967-967.132-r1-0-0
Degree $1$
Conductor $967$
Sign $0.995 + 0.0908i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.549 + 0.835i)2-s + (−0.987 − 0.155i)3-s + (−0.395 + 0.918i)4-s + (0.927 − 0.374i)5-s + (−0.413 − 0.910i)6-s + (0.837 − 0.547i)7-s + (−0.984 + 0.174i)8-s + (0.951 + 0.307i)9-s + (0.822 + 0.568i)10-s + (0.0876 − 0.996i)11-s + (0.533 − 0.845i)12-s + (0.906 + 0.422i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.687 − 0.726i)16-s + (−0.864 − 0.502i)17-s + ⋯
L(s)  = 1  + (0.549 + 0.835i)2-s + (−0.987 − 0.155i)3-s + (−0.395 + 0.918i)4-s + (0.927 − 0.374i)5-s + (−0.413 − 0.910i)6-s + (0.837 − 0.547i)7-s + (−0.984 + 0.174i)8-s + (0.951 + 0.307i)9-s + (0.822 + 0.568i)10-s + (0.0876 − 0.996i)11-s + (0.533 − 0.845i)12-s + (0.906 + 0.422i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.687 − 0.726i)16-s + (−0.864 − 0.502i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.995 + 0.0908i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.995 + 0.0908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.860511972 + 0.1301433019i\)
\(L(\frac12)\) \(\approx\) \(2.860511972 + 0.1301433019i\)
\(L(1)\) \(\approx\) \(1.373499882 + 0.3231157299i\)
\(L(1)\) \(\approx\) \(1.373499882 + 0.3231157299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.549 + 0.835i)T \)
3 \( 1 + (-0.987 - 0.155i)T \)
5 \( 1 + (0.927 - 0.374i)T \)
7 \( 1 + (0.837 - 0.547i)T \)
11 \( 1 + (0.0876 - 0.996i)T \)
13 \( 1 + (0.906 + 0.422i)T \)
17 \( 1 + (-0.864 - 0.502i)T \)
19 \( 1 + (0.953 + 0.300i)T \)
23 \( 1 + (0.844 - 0.536i)T \)
29 \( 1 + (-0.353 + 0.935i)T \)
31 \( 1 + (0.840 + 0.541i)T \)
37 \( 1 + (0.235 + 0.971i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (0.997 + 0.0714i)T \)
47 \( 1 + (0.430 - 0.902i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (0.190 - 0.981i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (-0.0487 - 0.998i)T \)
71 \( 1 + (-0.998 - 0.0585i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (-0.254 - 0.967i)T \)
83 \( 1 + (-0.979 + 0.200i)T \)
89 \( 1 + (0.889 + 0.457i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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

−21.50154715506248899047918885103, −20.98488767024096282522706913, −20.35484736117353126128471060835, −19.05138017953875079565398554617, −18.26957906475046851482684945610, −17.68611407108700076858058209705, −17.31162007674055291556424340479, −15.63884165692441965894542130806, −15.240218153950508448708560887412, −14.29685054822189931695073782093, −13.2789741896178132807284635792, −12.80544852060934561373778174683, −11.67075231135140091890246881838, −11.22961817661633956414702904687, −10.45354321089697337793167263501, −9.66445855630737642240025413391, −8.92607614936505838076523690487, −7.36889228340591903837563381, −6.23504939398466641559322497155, −5.68942801098876692146505865081, −4.88817357056132309296426438844, −4.10851071688975083930356971691, −2.74450915372709598743520274749, −1.7822126749699600081691853271, −1.03926437991382226401567676440, 0.66877436125924641322824710550, 1.591861469157347889860231273, 3.17504544300819541323352366810, 4.45712601288912809298548921791, 5.01570918908877543343671188717, 5.83877159832294285592390925814, 6.54072465239874098241403432608, 7.31332125251230669823596877040, 8.47159901350167391325587493127, 9.12552642767822985929541946997, 10.45855694635732503631105644185, 11.23454354084621874851951517558, 11.98363584389679453997931711753, 13.0400927504824670625576754843, 13.74013548125084997953158024688, 14.0602240619670516246272034610, 15.38660778929045406067573372511, 16.30299790843104538956417396948, 16.71973828337199921835624674059, 17.46740450508571300944316855561, 18.13085612258536274851172244829, 18.71217798125956500729240318625, 20.49329721549554411263855928345, 20.95517156407638724441023321098, 21.83231341652015976217269903605

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