↑

Properties

Label 2-297-99.29-c1-0-8
Degree $2$
Conductor $297$
Sign $0.466 + 0.884i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
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  + (1.65 − 0.738i)2-s + (0.864 − 0.960i)4-s + (0.488 − 1.09i)5-s + (1.00 − 4.73i)7-s + (−0.396 + 1.22i)8-s − 2.18i·10-s + (−1.36 + 3.02i)11-s + (3.04 + 0.319i)13-s + (−1.82 − 8.59i)14-s + (0.513 + 4.88i)16-s + (1.80 − 1.31i)17-s + (−4.34 − 1.41i)19-s + (−0.631 − 1.41i)20-s + (−0.0397 + 6.01i)22-s + (−1.72 + 0.993i)23-s + ⋯
L(s)  = 1  + (1.17 − 0.521i)2-s + (0.432 − 0.480i)4-s + (0.218 − 0.490i)5-s + (0.380 − 1.78i)7-s + (−0.140 + 0.431i)8-s − 0.689i·10-s + (−0.412 + 0.910i)11-s + (0.844 + 0.0887i)13-s + (−0.488 − 2.29i)14-s + (0.128 + 1.22i)16-s + (0.438 − 0.318i)17-s + (−0.996 − 0.323i)19-s + (−0.141 − 0.317i)20-s + (−0.00847 + 1.28i)22-s + (−0.358 + 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99308 - 1.20254i\)
\(L(\frac12)\) \(\approx\) \(1.99308 - 1.20254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (1.36 - 3.02i)T \)
good2 \( 1 + (-1.65 + 0.738i)T + (1.33 - 1.48i)T^{2} \)
5 \( 1 + (-0.488 + 1.09i)T + (-3.34 - 3.71i)T^{2} \)
7 \( 1 + (-1.00 + 4.73i)T + (-6.39 - 2.84i)T^{2} \)
13 \( 1 + (-3.04 - 0.319i)T + (12.7 + 2.70i)T^{2} \)
17 \( 1 + (-1.80 + 1.31i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.34 + 1.41i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.72 - 0.993i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.65 - 0.352i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (0.555 - 5.28i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-0.344 - 1.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (10.3 - 2.21i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-3.28 - 1.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.42 - 2.18i)T + (4.91 - 46.7i)T^{2} \)
53 \( 1 + (0.749 - 1.03i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0234 + 0.0211i)T + (6.16 + 58.6i)T^{2} \)
61 \( 1 + (-1.18 + 0.124i)T + (59.6 - 12.6i)T^{2} \)
67 \( 1 + (-1.05 - 1.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.26 + 3.11i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.69 + 2.17i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.55 + 3.49i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (0.759 + 7.22i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + (-13.2 + 5.89i)T + (64.9 - 72.0i)T^{2} \)
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

−11.71346877451289321185213319528, −10.79858804420494656633684993493, −10.14851701532124491167019652239, −8.708113231438591065671457299518, −7.63670956499410211791074667468, −6.52312260745564742279898073132, −5.03960808504752780594094488987, −4.42742501692517692960820858341, −3.38729631892779835117415931115, −1.58276652922349435573802083132, 2.44947041177587453363440771405, 3.65418401435992745860628594027, 5.11712503079721817474989119739, 5.93397270919392306970667818888, 6.42902563598164133673110923898, 8.124814273697273419182027962494, 8.831178333329388930944558158712, 10.15859772169757595897517527701, 11.24878490476110400474913405065, 12.17810794067053246185852214604

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