↑

Properties

Label 2-722-361.144-c1-0-24
Degree $2$
Conductor $722$
Sign $-0.616 + 0.787i$
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  + (0.851 − 0.523i)2-s + (−1.09 + 2.93i)3-s + (0.451 − 0.892i)4-s + (−0.462 − 0.629i)5-s + (0.600 + 3.07i)6-s + (−0.692 + 1.06i)7-s + (−0.0825 − 0.996i)8-s + (−5.13 − 4.47i)9-s + (−0.724 − 0.294i)10-s + (−3.67 − 1.26i)11-s + (2.12 + 2.30i)12-s + (−4.36 − 5.29i)13-s + (−0.0348 + 1.26i)14-s + (2.35 − 0.666i)15-s + (−0.592 − 0.805i)16-s + (−0.696 − 1.37i)17-s + ⋯
L(s)  = 1  + (0.602 − 0.370i)2-s + (−0.634 + 1.69i)3-s + (0.225 − 0.446i)4-s + (−0.207 − 0.281i)5-s + (0.245 + 1.25i)6-s + (−0.261 + 0.400i)7-s + (−0.0291 − 0.352i)8-s + (−1.71 − 1.49i)9-s + (−0.229 − 0.0930i)10-s + (−1.10 − 0.380i)11-s + (0.612 + 0.665i)12-s + (−1.21 − 1.46i)13-s + (−0.00932 + 0.338i)14-s + (0.608 − 0.172i)15-s + (−0.148 − 0.201i)16-s + (−0.168 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.616 + 0.787i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104977 - 0.215578i\)
\(L(\frac12)\) \(\approx\) \(0.104977 - 0.215578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.851 + 0.523i)T \)
19 \( 1 + (-2.87 - 3.27i)T \)
good3 \( 1 + (1.09 - 2.93i)T + (-2.26 - 1.97i)T^{2} \)
5 \( 1 + (0.462 + 0.629i)T + (-1.49 + 4.77i)T^{2} \)
7 \( 1 + (0.692 - 1.06i)T + (-2.81 - 6.41i)T^{2} \)
11 \( 1 + (3.67 + 1.26i)T + (8.68 + 6.75i)T^{2} \)
13 \( 1 + (4.36 + 5.29i)T + (-2.49 + 12.7i)T^{2} \)
17 \( 1 + (0.696 + 1.37i)T + (-10.0 + 13.6i)T^{2} \)
23 \( 1 + (-1.43 - 3.84i)T + (-17.3 + 15.1i)T^{2} \)
29 \( 1 + (-2.70 - 0.149i)T + (28.8 + 3.19i)T^{2} \)
31 \( 1 + (5.57 - 3.01i)T + (16.9 - 25.9i)T^{2} \)
37 \( 1 + (1.99 + 0.684i)T + (29.1 + 22.7i)T^{2} \)
41 \( 1 + (6.74 + 6.55i)T + (1.12 + 40.9i)T^{2} \)
43 \( 1 + (7.40 - 3.00i)T + (30.8 - 29.9i)T^{2} \)
47 \( 1 + (1.36 + 7.01i)T + (-43.5 + 17.6i)T^{2} \)
53 \( 1 + (-2.48 - 12.7i)T + (-49.1 + 19.9i)T^{2} \)
59 \( 1 + (6.94 + 6.75i)T + (1.62 + 58.9i)T^{2} \)
61 \( 1 + (-7.96 + 3.75i)T + (38.7 - 47.0i)T^{2} \)
67 \( 1 + (9.56 - 6.63i)T + (23.4 - 62.7i)T^{2} \)
71 \( 1 + (10.9 + 5.17i)T + (45.1 + 54.8i)T^{2} \)
73 \( 1 + (2.61 + 5.16i)T + (-43.2 + 58.8i)T^{2} \)
79 \( 1 + (-8.34 + 3.38i)T + (56.6 - 55.0i)T^{2} \)
83 \( 1 + (2.82 + 6.43i)T + (-56.2 + 61.0i)T^{2} \)
89 \( 1 + (3.98 - 7.86i)T + (-52.7 - 71.7i)T^{2} \)
97 \( 1 + (-2.35 - 1.63i)T + (34.0 + 90.8i)T^{2} \)
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

−10.38558441435230175588333915574, −9.620781288950212617878921756175, −8.660212708272843000187606893444, −7.50393906688970013325480731940, −5.92551754340496464034686780586, −5.22568026862490216738912183609, −4.87700991078657040204888631527, −3.50522183096161329483744520562, −2.86222476843796540770335034176, −0.10132189451478400911662959409, 1.89344667604087051971157847983, 2.91393330712955246783044212080, 4.65113710208658528630910358788, 5.42187295220733073212915545125, 6.66693820149176087709073654752, 7.03082941456949303452211698684, 7.58282524239646245706340709643, 8.629708021627278726872534629354, 10.00007465406101229069491976579, 11.14650791295569567445615557044

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