↑

Properties

Label 2-20-20.3-c9-0-14
Degree $2$
Conductor $20$
Sign $0.0770 - 0.997i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (13.1 + 18.4i)2-s + (30.0 − 30.0i)3-s + (−166. + 484. i)4-s + (965. − 1.01e3i)5-s + (948. + 158. i)6-s + (6.80e3 + 6.80e3i)7-s + (−1.11e4 + 3.28e3i)8-s + 1.78e4i·9-s + (3.13e4 + 4.50e3i)10-s − 1.99e4i·11-s + (9.53e3 + 1.95e4i)12-s + (2.93e4 + 2.93e4i)13-s + (−3.59e4 + 2.14e5i)14-s + (−1.36e3 − 5.93e4i)15-s + (−2.06e5 − 1.61e5i)16-s + (−3.98e5 + 3.98e5i)17-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (0.214 − 0.214i)3-s + (−0.325 + 0.945i)4-s + (0.690 − 0.723i)5-s + (0.298 + 0.0500i)6-s + (1.07 + 1.07i)7-s + (−0.958 + 0.283i)8-s + 0.908i·9-s + (0.989 + 0.142i)10-s − 0.410i·11-s + (0.132 + 0.272i)12-s + (0.285 + 0.285i)13-s + (−0.250 + 1.49i)14-s + (−0.00698 − 0.302i)15-s + (−0.787 − 0.616i)16-s + (−1.15 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0770 - 0.997i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0770 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.0770 - 0.997i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ 0.0770 - 0.997i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.06693 + 1.91332i\)
\(L(\frac12)\) \(\approx\) \(2.06693 + 1.91332i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-13.1 - 18.4i)T \)
5 \( 1 + (-965. + 1.01e3i)T \)
good3 \( 1 + (-30.0 + 30.0i)T - 1.96e4iT^{2} \)
7 \( 1 + (-6.80e3 - 6.80e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 1.99e4iT - 2.35e9T^{2} \)
13 \( 1 + (-2.93e4 - 2.93e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (3.98e5 - 3.98e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 7.44e5T + 3.22e11T^{2} \)
23 \( 1 + (-6.85e5 + 6.85e5i)T - 1.80e12iT^{2} \)
29 \( 1 + 8.75e5iT - 1.45e13T^{2} \)
31 \( 1 + 9.02e6iT - 2.64e13T^{2} \)
37 \( 1 + (-8.16e6 + 8.16e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.35e7T + 3.27e14T^{2} \)
43 \( 1 + (5.41e5 - 5.41e5i)T - 5.02e14iT^{2} \)
47 \( 1 + (-2.77e6 - 2.77e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (3.31e7 + 3.31e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 8.12e7T + 8.66e15T^{2} \)
61 \( 1 - 3.54e7T + 1.16e16T^{2} \)
67 \( 1 + (5.83e7 + 5.83e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.92e8iT - 4.58e16T^{2} \)
73 \( 1 + (8.87e7 + 8.87e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 9.69e7T + 1.19e17T^{2} \)
83 \( 1 + (-4.71e8 + 4.71e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 4.03e8iT - 3.50e17T^{2} \)
97 \( 1 + (-9.57e8 + 9.57e8i)T - 7.60e17iT^{2} \)
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

−16.41148999299562116915335884586, −15.12270513570983091120459968412, −13.87189386053030106824650295058, −12.95295639790235148889627717069, −11.45167366092323217671742563986, −8.941623013959435170538501623049, −8.039620495956627502644430876991, −5.91842764878007207609541692849, −4.76919059871469782066712787139, −2.11361745078266747070482165546, 1.26943983863734358203236674559, 3.19194807498490480214388682957, 4.90231682423550296800362390082, 6.90423886415265969428427779881, 9.355129443605468455729886898067, 10.57155489024127597090127689458, 11.65369231131194705293767331778, 13.52972440416033282317759081967, 14.24754959968225754767434446158, 15.37148957679767688533435647552

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