↑

Properties

Label 2-300-300.71-c1-0-18
Degree $2$
Conductor $300$
Sign $0.838 - 0.545i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (1.35 − 0.398i)2-s + (−1.70 + 0.329i)3-s + (1.68 − 1.08i)4-s + (0.213 + 2.22i)5-s + (−2.17 + 1.12i)6-s + 2.98i·7-s + (1.85 − 2.13i)8-s + (2.78 − 1.12i)9-s + (1.17 + 2.93i)10-s + (0.156 − 0.482i)11-s + (−2.50 + 2.39i)12-s + (0.990 + 3.04i)13-s + (1.19 + 4.05i)14-s + (−1.09 − 3.71i)15-s + (1.66 − 3.63i)16-s + (−1.17 − 1.61i)17-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (−0.981 + 0.190i)3-s + (0.841 − 0.540i)4-s + (0.0952 + 0.995i)5-s + (−0.888 + 0.459i)6-s + 1.12i·7-s + (0.654 − 0.755i)8-s + (0.927 − 0.374i)9-s + (0.371 + 0.928i)10-s + (0.0472 − 0.145i)11-s + (−0.722 + 0.690i)12-s + (0.274 + 0.845i)13-s + (0.318 + 1.08i)14-s + (−0.283 − 0.959i)15-s + (0.415 − 0.909i)16-s + (−0.284 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.838 - 0.545i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.838 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69779 + 0.503589i\)
\(L(\frac12)\) \(\approx\) \(1.69779 + 0.503589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.398i)T \)
3 \( 1 + (1.70 - 0.329i)T \)
5 \( 1 + (-0.213 - 2.22i)T \)
good7 \( 1 - 2.98iT - 7T^{2} \)
11 \( 1 + (-0.156 + 0.482i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.990 - 3.04i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.17 + 1.61i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.79 - 5.22i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.913 + 2.81i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.48 + 7.54i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.51 + 6.21i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.55 + 4.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (7.57 - 2.46i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.65iT - 43T^{2} \)
47 \( 1 + (1.98 + 1.44i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.87 - 8.08i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.00 + 3.08i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.34 + 10.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.33 - 3.21i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.52 + 4.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.85 + 5.70i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.47 + 6.16i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.94 + 7.22i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.844 - 0.274i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.04 - 2.93i)T + (29.9 + 92.2i)T^{2} \)
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

−11.69899823634237734064541456436, −11.30020754912409149165449163009, −10.23827581256390418869094148553, −9.432041613441599393300485479828, −7.59587896560430986100272526501, −6.35313343459040877670453143746, −5.99453715519596355310688328296, −4.79079200665235302840539855860, −3.52101925237156140907334748133, −2.09062989351166068101386597876, 1.27277515426779953419788724210, 3.58868084335281114529740283937, 4.85950656854804744059139343120, 5.33293980002924806885333396271, 6.70533993158350296185280667406, 7.37310455785807487105500251829, 8.583574097830532688396344421149, 10.17344062371890191290066942301, 10.92651336881786783848369216820, 11.86508644470171880001265730290

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