↑

Properties

Label 2-21e2-7.2-c3-0-8
Degree $2$
Conductor $441$
Sign $-0.266 + 0.963i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
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  + (−2.63 + 4.56i)2-s + (−9.91 − 17.1i)4-s + (−5.27 + 9.13i)5-s + 62.3·8-s + (−27.8 − 48.1i)10-s + (17.3 + 30.0i)11-s + 37.2·13-s + (−85.2 + 147. i)16-s + (5.27 + 9.13i)17-s + (−29.2 + 50.7i)19-s + 209.·20-s − 183.·22-s + (−62.6 + 108. i)23-s + (6.85 + 11.8i)25-s + (−98.3 + 170. i)26-s + ⋯
L(s)  = 1  + (−0.932 + 1.61i)2-s + (−1.23 − 2.14i)4-s + (−0.471 + 0.817i)5-s + 2.75·8-s + (−0.879 − 1.52i)10-s + (0.476 + 0.824i)11-s + 0.795·13-s + (−1.33 + 2.30i)16-s + (0.0752 + 0.130i)17-s + (−0.353 + 0.612i)19-s + 2.33·20-s − 1.77·22-s + (−0.568 + 0.984i)23-s + (0.0548 + 0.0949i)25-s + (−0.742 + 1.28i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5932428297\)
\(L(\frac12)\) \(\approx\) \(0.5932428297\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (2.63 - 4.56i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (5.27 - 9.13i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-17.3 - 30.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 37.2T + 2.19e3T^{2} \)
17 \( 1 + (-5.27 - 9.13i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (29.2 - 50.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (62.6 - 108. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 35.4T + 2.43e4T^{2} \)
31 \( 1 + (-145. - 252. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-129. + 225. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 338.T + 6.89e4T^{2} \)
43 \( 1 - 6.80T + 7.95e4T^{2} \)
47 \( 1 + (125. - 217. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (268. + 464. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-17.9 - 31.0i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-28.8 + 50.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (240. + 417. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 363.T + 3.57e5T^{2} \)
73 \( 1 + (-290. - 503. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-346. + 600. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + (-176. + 305. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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

−10.94080457936104133041017135119, −10.15888017925232609449094393856, −9.342345248846275533346159520721, −8.364469967994543348277806729338, −7.64135241977008896837526208929, −6.79292005794462095563738077326, −6.18576822211623830756651705686, −4.98141965968518431136609820929, −3.67886185264086729964300817287, −1.47285031319865108133670216426, 0.32053462587336742963830725481, 1.20381379075134864765827905941, 2.65673099837421436977593606398, 3.82306954167273412894877475754, 4.63618303678851795812543948322, 6.38140146775335437183001230643, 8.042277825568201578768308346523, 8.464172505288079405517787288773, 9.198764274960703586896977527931, 10.15431931500376265880680011674

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