↑

Properties

Label 2-21-7.4-c5-0-4
Degree $2$
Conductor $21$
Sign $0.857 + 0.514i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (4.5 − 7.79i)3-s + (14 − 24.2i)4-s + (−5.5 − 9.52i)5-s + 18·6-s + (129.5 − 6.06i)7-s + 120·8-s + (−40.5 − 70.1i)9-s + (11 − 19.0i)10-s + (−134.5 + 232. i)11-s + (−126 − 218. i)12-s − 308·13-s + (140 + 218. i)14-s − 99·15-s + (−328 − 568. i)16-s + (−948 + 1.64e3i)17-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (0.288 − 0.499i)3-s + (0.437 − 0.757i)4-s + (−0.0983 − 0.170i)5-s + 0.204·6-s + (0.998 − 0.0467i)7-s + 0.662·8-s + (−0.166 − 0.288i)9-s + (0.0347 − 0.0602i)10-s + (−0.335 + 0.580i)11-s + (−0.252 − 0.437i)12-s − 0.505·13-s + (0.190 + 0.297i)14-s − 0.113·15-s + (−0.320 − 0.554i)16-s + (−0.795 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.77862 - 0.492437i\)
\(L(\frac12)\) \(\approx\) \(1.77862 - 0.492437i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 + (-129.5 + 6.06i)T \)
good2 \( 1 + (-1 - 1.73i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (5.5 + 9.52i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (134.5 - 232. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 308T + 3.71e5T^{2} \)
17 \( 1 + (948 - 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-82 - 142. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.63e3 - 2.82e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.41e3T + 2.05e7T^{2} \)
31 \( 1 + (1.42e3 - 2.46e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.66e3 - 9.81e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.68e4T + 1.15e8T^{2} \)
43 \( 1 + 7.89e3T + 1.47e8T^{2} \)
47 \( 1 + (1.05e4 + 1.82e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.48e4 + 2.57e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-4.08e3 + 7.06e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (7.58e3 + 1.31e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.60e4 + 2.77e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.82e4T + 1.80e9T^{2} \)
73 \( 1 + (1.74e4 - 3.01e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (6.76e3 + 1.17e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.81e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.74e4 - 9.95e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.54e5T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.05913613541998908204235982049, −15.34398538378447585864942778982, −14.64668832105816462542178637351, −13.28487970260665828998566639764, −11.68393884921847301776416933191, −10.26754123727480900395552315345, −8.270496920005879449165935584238, −6.81750658270653134090017068062, −4.97554966483568235792382431637, −1.75412286179518563771849019185, 2.72825060340540242553584079017, 4.69903887003035454673668142909, 7.35144283289989612574614320361, 8.754702276809893504435825252344, 10.78221884022388611520688050872, 11.69275279578636196646526949339, 13.32028728743162366899138785025, 14.67068236937261504725333614358, 15.96637152045561049916619260318, 17.07992254346787361207996444366

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