Properties

Label 2-300-300.287-c2-0-89
Degree $2$
Conductor $300$
Sign $0.994 + 0.103i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 + 0.151i)2-s + (−1.11 + 2.78i)3-s + (3.95 + 0.606i)4-s + (0.179 − 4.99i)5-s + (−2.65 + 5.38i)6-s + (8.22 − 8.22i)7-s + (7.79 + 1.80i)8-s + (−6.49 − 6.22i)9-s + (1.11 − 9.93i)10-s + (−1.28 − 0.935i)11-s + (−6.11 + 10.3i)12-s + (2.67 − 16.8i)13-s + (17.6 − 15.1i)14-s + (13.7 + 6.09i)15-s + (15.2 + 4.79i)16-s + (11.2 + 22.1i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0759i)2-s + (−0.372 + 0.927i)3-s + (0.988 + 0.151i)4-s + (0.0359 − 0.999i)5-s + (−0.442 + 0.896i)6-s + (1.17 − 1.17i)7-s + (0.974 + 0.226i)8-s + (−0.721 − 0.692i)9-s + (0.111 − 0.993i)10-s + (−0.117 − 0.0850i)11-s + (−0.509 + 0.860i)12-s + (0.205 − 1.29i)13-s + (1.26 − 1.08i)14-s + (0.913 + 0.406i)15-s + (0.954 + 0.299i)16-s + (0.663 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.994 + 0.103i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.98874 - 0.154785i\)
\(L(\frac12)\) \(\approx\) \(2.98874 - 0.154785i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 - 0.151i)T \)
3 \( 1 + (1.11 - 2.78i)T \)
5 \( 1 + (-0.179 + 4.99i)T \)
good7 \( 1 + (-8.22 + 8.22i)T - 49iT^{2} \)
11 \( 1 + (1.28 + 0.935i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (-2.67 + 16.8i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-11.2 - 22.1i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (3.50 - 10.7i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (31.9 - 5.05i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (-10.1 - 31.3i)T + (-680. + 494. i)T^{2} \)
31 \( 1 + (-10.4 - 3.38i)T + (777. + 564. i)T^{2} \)
37 \( 1 + (-32.8 - 5.21i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (10.7 + 14.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + (14.1 + 14.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (33.4 - 65.6i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (-9.37 + 18.3i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (31.1 + 42.8i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-63.9 - 46.4i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-31.0 - 60.9i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (26.1 + 80.5i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (127. - 20.1i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (32.1 + 98.9i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-32.0 - 62.9i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (2.59 + 1.88i)T + (2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (15.6 - 30.7i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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

−11.57392795506449935816505717209, −10.54331366445405051049634583336, −10.17573967473566112165110251958, −8.312459255862454324882810500365, −7.83849366940613043247887422278, −6.05560306564848200764253560505, −5.26904634289215719905669641929, −4.37339678411813729501940253238, −3.59778189226674020213577214381, −1.31541148449420036790991110421, 1.92209387905623436422351736750, 2.67320872172313172841233369170, 4.54376917895912795015375645678, 5.64686659112742673165265842431, 6.45353840591968727970503108871, 7.39013110188788686869458294316, 8.302936159363623142790801487480, 9.921678437138485829719158999629, 11.40651457154269555525813120288, 11.50591141835674923495208180522

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