↑

Properties

Label 2-242-11.3-c3-0-20
Degree $2$
Conductor $242$
Sign $-0.394 + 0.918i$
Analytic cond. $14.2784$
Root an. cond. $3.77868$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 1.17i)2-s + (−2.16 − 6.65i)3-s + (1.23 − 3.80i)4-s + (15.3 + 11.1i)5-s + (−11.3 − 8.22i)6-s + (4.32 − 13.3i)7-s + (−2.47 − 7.60i)8-s + (−17.7 + 12.9i)9-s + 38·10-s − 28·12-s + (58.2 − 42.3i)13-s + (−8.65 − 26.6i)14-s + (41.0 − 126. i)15-s + (−12.9 − 9.40i)16-s + (37.2 + 27.0i)17-s + (−13.5 + 41.8i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.416 − 1.28i)3-s + (0.154 − 0.475i)4-s + (1.37 + 0.998i)5-s + (−0.770 − 0.559i)6-s + (0.233 − 0.718i)7-s + (−0.109 − 0.336i)8-s + (−0.659 + 0.478i)9-s + 1.20·10-s − 0.673·12-s + (1.24 − 0.902i)13-s + (−0.165 − 0.508i)14-s + (0.707 − 2.17i)15-s + (−0.202 − 0.146i)16-s + (0.530 + 0.385i)17-s + (−0.178 + 0.547i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(14.2784\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :3/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55289 - 2.35703i\)
\(L(\frac12)\) \(\approx\) \(1.55289 - 2.35703i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 + 1.17i)T \)
11 \( 1 \)
good3 \( 1 + (2.16 + 6.65i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (-15.3 - 11.1i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-4.32 + 13.3i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-58.2 + 42.3i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-37.2 - 27.0i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (6.18 + 19.0i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + 107T + 1.21e4T^{2} \)
29 \( 1 + (-37.0 + 114. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (94.6 - 68.7i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (62.1 - 191. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (70.4 + 216. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 242T + 7.95e4T^{2} \)
47 \( 1 + (29.6 + 91.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (370. - 269. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-134. + 413. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-540. - 392. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 439T + 3.00e5T^{2} \)
71 \( 1 + (-900. - 654. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (22.2 - 68.4i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-56.6 + 41.1i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (289. + 210. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 895T + 7.04e5T^{2} \)
97 \( 1 + (330. - 240. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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

−11.37742861560736580517129011414, −10.56955268955982750079934958538, −9.910640212721398422804653004878, −8.157713138005833121128117569675, −6.96763576051939353725265584257, −6.23549125188950712254349521776, −5.52980651182890151205015686083, −3.54861781281908589959356630401, −2.13455761894216398315631848491, −1.10125095059929577839180841503, 1.83966700925145580868575068189, 3.78545497740369775409294383016, 4.94434783191715792275016143541, 5.54757376074780306613960118372, 6.37493802621862305781316296310, 8.358637665896907758931170667775, 9.193212788271716430589019363464, 9.867754653821023658793287205108, 11.03019996803192117011108652570, 12.00796456598079621039587276149

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