↑

Properties

Label 2-722-19.5-c1-0-14
Degree $2$
Conductor $722$
Sign $0.973 + 0.226i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (2 + 3.46i)7-s + (0.500 − 0.866i)8-s + (1.87 + 0.684i)9-s + (−1.5 + 2.59i)11-s + (−0.5 − 0.866i)12-s + (0.347 + 1.96i)13-s + (3.06 + 2.57i)14-s + (0.173 − 0.984i)16-s + (5.63 − 2.05i)17-s + 2·18-s + (3.75 − 1.36i)21-s + (−0.520 + 2.95i)22-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.100 − 0.568i)3-s + (0.383 − 0.321i)4-s + (−0.0708 − 0.402i)6-s + (0.755 + 1.30i)7-s + (0.176 − 0.306i)8-s + (0.626 + 0.228i)9-s + (−0.452 + 0.783i)11-s + (−0.144 − 0.249i)12-s + (0.0963 + 0.546i)13-s + (0.818 + 0.687i)14-s + (0.0434 − 0.246i)16-s + (1.36 − 0.497i)17-s + 0.471·18-s + (0.820 − 0.298i)21-s + (−0.111 + 0.629i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.973 + 0.226i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.973 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58273 - 0.296576i\)
\(L(\frac12)\) \(\approx\) \(2.58273 - 0.296576i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.347 - 1.96i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-5.63 + 2.05i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.59 - 3.85i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-1.56 + 8.86i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.06 + 2.57i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.59 + 3.85i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-8.45 + 3.07i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.06 - 2.57i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-6.57 - 2.39i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.59 + 3.85i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.173 - 0.984i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.694 - 3.93i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.04 - 5.90i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (15.9 - 5.81i)T + (74.3 - 62.3i)T^{2} \)
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

−10.36408914547526596631211135457, −9.695886448305372882660010501033, −8.510085883919053998091877992127, −7.67881359231139706893836035473, −6.88999569306050725163235362429, −5.65375093297046826744156728378, −5.07509753623535944850096342668, −3.90504931861227729387431166062, −2.38506236960954075175712581519, −1.72588733642421442884050129811, 1.31984503617888689821392152302, 3.28096676694320416899684491907, 3.95005449273002420896842129601, 4.90281160968276419407991246234, 5.77822236012031607840519395761, 6.98445401061566677435992171486, 7.78408397157573197355093402611, 8.457244559408501198271998069614, 9.921101143569989171319260545039, 10.45557724226907285696699260733

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