↑

Properties

Label 2-30e2-36.11-c1-0-32
Degree $2$
Conductor $900$
Sign $0.941 - 0.336i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (0.282 − 1.38i)2-s + (1.44 + 0.959i)3-s + (−1.84 − 0.782i)4-s + (1.73 − 1.72i)6-s + (0.953 + 0.550i)7-s + (−1.60 + 2.32i)8-s + (1.15 + 2.76i)9-s + (−2.84 + 4.92i)11-s + (−1.90 − 2.89i)12-s + (1.19 + 2.07i)13-s + (1.03 − 1.16i)14-s + (2.77 + 2.88i)16-s + 1.29i·17-s + (4.16 − 0.822i)18-s − 2.78i·19-s + ⋯
L(s)  = 1  + (0.199 − 0.979i)2-s + (0.832 + 0.554i)3-s + (−0.920 − 0.391i)4-s + (0.709 − 0.705i)6-s + (0.360 + 0.208i)7-s + (−0.567 + 0.823i)8-s + (0.385 + 0.922i)9-s + (−0.856 + 1.48i)11-s + (−0.549 − 0.835i)12-s + (0.332 + 0.575i)13-s + (0.275 − 0.311i)14-s + (0.693 + 0.720i)16-s + 0.313i·17-s + (0.981 − 0.193i)18-s − 0.638i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.941 - 0.336i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.941 - 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92688 + 0.333889i\)
\(L(\frac12)\) \(\approx\) \(1.92688 + 0.333889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.282 + 1.38i)T \)
3 \( 1 + (-1.44 - 0.959i)T \)
5 \( 1 \)
good7 \( 1 + (-0.953 - 0.550i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.84 - 4.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.19 - 2.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.29iT - 17T^{2} \)
19 \( 1 + 2.78iT - 19T^{2} \)
23 \( 1 + (-1.05 - 1.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.51 - 3.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.78 + 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 + (-6.35 + 3.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.90 + 2.25i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.67 - 8.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (-3.66 - 6.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.48 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.20 - 1.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 9.86T + 73T^{2} \)
79 \( 1 + (-0.975 - 0.563i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.76 - 9.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.16iT - 89T^{2} \)
97 \( 1 + (-6.42 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.12442276280495965159837964577, −9.517509347133208171398822994464, −8.682059979213009757427455788045, −7.972983794974907930433199358489, −6.84465061952463099466189776190, −5.19020663472102112393922172601, −4.70854563368357630970388819729, −3.72314397739100059254960301298, −2.58449362690729870991960220103, −1.77856922319448941708799597171, 0.833196102283733957713039489415, 2.82671257047335363618798974094, 3.63321702878495283380814436696, 4.88544917398796416399945749276, 5.91293303494454121512569161499, 6.64567628697718310192726672007, 7.76649786345858852247364859081, 8.194652853395597290172416570553, 8.756327971257093517045923141257, 9.846270249365274325026667512455

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