↑

Properties

Label 2-605-55.2-c1-0-18
Degree $2$
Conductor $605$
Sign $-0.506 - 0.862i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (0.690 + 1.35i)2-s + (−0.124 + 0.787i)3-s + (−0.182 + 0.251i)4-s + (0.543 + 2.16i)5-s + (−1.15 + 0.374i)6-s + (−0.189 + 0.0299i)7-s + (2.53 + 0.401i)8-s + (2.24 + 0.730i)9-s + (−2.56 + 2.23i)10-s + (−0.175 − 0.175i)12-s + (2.75 − 1.40i)13-s + (−0.171 − 0.235i)14-s + (−1.77 + 0.157i)15-s + (1.39 + 4.30i)16-s + (−3.30 − 1.68i)17-s + (0.562 + 3.54i)18-s + ⋯
L(s)  = 1  + (0.487 + 0.957i)2-s + (−0.0720 + 0.454i)3-s + (−0.0912 + 0.125i)4-s + (0.243 + 0.969i)5-s + (−0.470 + 0.152i)6-s + (−0.0714 + 0.0113i)7-s + (0.896 + 0.142i)8-s + (0.749 + 0.243i)9-s + (−0.810 + 0.706i)10-s + (−0.0505 − 0.0505i)12-s + (0.764 − 0.389i)13-s + (−0.0457 − 0.0629i)14-s + (−0.458 + 0.0407i)15-s + (0.349 + 1.07i)16-s + (−0.800 − 0.408i)17-s + (0.132 + 0.836i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.506 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10325 + 1.92702i\)
\(L(\frac12)\) \(\approx\) \(1.10325 + 1.92702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.543 - 2.16i)T \)
11 \( 1 \)
good2 \( 1 + (-0.690 - 1.35i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (0.124 - 0.787i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (0.189 - 0.0299i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.75 + 1.40i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (3.30 + 1.68i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.601 + 0.437i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.14 - 1.14i)T - 23iT^{2} \)
29 \( 1 + (7.72 + 5.61i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.108 + 0.333i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.845 - 5.33i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (3.93 + 5.41i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.72 - 3.72i)T + 43iT^{2} \)
47 \( 1 + (-12.2 - 1.93i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (4.09 + 8.04i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-5.65 + 7.78i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.60 - 1.82i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \)
71 \( 1 + (3.54 + 10.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.375 - 2.37i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.207 + 0.637i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.57 + 14.8i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (-1.22 + 0.626i)T + (57.0 - 78.4i)T^{2} \)
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

−10.85768924821074558330737575399, −10.16760530298723476437116779148, −9.308961779626473611232479970250, −7.914721896741135630755197180853, −7.22289058372319497505863356859, −6.38433347962382494462558031466, −5.64003383929309936110638839661, −4.56572599012410055518044664850, −3.57713004428299181251643976330, −1.97928969912883222857172896010, 1.24419879019669281212141578991, 2.12760122607726065513378661518, 3.76181718224217604482870803298, 4.39855299847176302551498649783, 5.62021505133285352886967122136, 6.76319000746188261092603193935, 7.68890938403317081374038256922, 8.765950869535552062094925626404, 9.579534563560550064655647303463, 10.61628254921245217716519871908

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