↑

Properties

Label 2-300-60.47-c2-0-64
Degree $2$
Conductor $300$
Sign $-0.728 + 0.685i$
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  + (1.75 − 0.961i)2-s + (−0.903 − 2.86i)3-s + (2.15 − 3.37i)4-s + (−4.33 − 4.14i)6-s + (7.30 − 7.30i)7-s + (0.535 − 7.98i)8-s + (−7.36 + 5.17i)9-s + 4.41·11-s + (−11.5 − 3.11i)12-s + (−7.53 + 7.53i)13-s + (5.78 − 19.8i)14-s + (−6.73 − 14.5i)16-s + (0.350 − 0.350i)17-s + (−7.94 + 16.1i)18-s + 9.24·19-s + ⋯
L(s)  = 1  + (0.876 − 0.480i)2-s + (−0.301 − 0.953i)3-s + (0.538 − 0.842i)4-s + (−0.722 − 0.691i)6-s + (1.04 − 1.04i)7-s + (0.0669 − 0.997i)8-s + (−0.818 + 0.574i)9-s + 0.401·11-s + (−0.965 − 0.259i)12-s + (−0.579 + 0.579i)13-s + (0.413 − 1.41i)14-s + (−0.420 − 0.907i)16-s + (0.0206 − 0.0206i)17-s + (−0.441 + 0.897i)18-s + 0.486·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.728 + 0.685i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.728 + 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.941063 - 2.37317i\)
\(L(\frac12)\) \(\approx\) \(0.941063 - 2.37317i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 + 0.961i)T \)
3 \( 1 + (0.903 + 2.86i)T \)
5 \( 1 \)
good7 \( 1 + (-7.30 + 7.30i)T - 49iT^{2} \)
11 \( 1 - 4.41T + 121T^{2} \)
13 \( 1 + (7.53 - 7.53i)T - 169iT^{2} \)
17 \( 1 + (-0.350 + 0.350i)T - 289iT^{2} \)
19 \( 1 - 9.24T + 361T^{2} \)
23 \( 1 + (17.9 - 17.9i)T - 529iT^{2} \)
29 \( 1 - 5.52T + 841T^{2} \)
31 \( 1 + 48.1iT - 961T^{2} \)
37 \( 1 + (3.39 + 3.39i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.0iT - 1.68e3T^{2} \)
43 \( 1 + (1.45 + 1.45i)T + 1.84e3iT^{2} \)
47 \( 1 + (-27.8 - 27.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-52.6 - 52.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 46.1T + 3.72e3T^{2} \)
67 \( 1 + (-32.1 + 32.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 116.T + 5.04e3T^{2} \)
73 \( 1 + (-72.2 + 72.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 55.9T + 6.24e3T^{2} \)
83 \( 1 + (-46.5 + 46.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 33.2T + 7.92e3T^{2} \)
97 \( 1 + (-24.6 - 24.6i)T + 9.40e3iT^{2} \)
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

−11.51245806138758740364522840012, −10.65029873933846052246068634748, −9.512493925881075141909928535953, −7.86295274073601549659519309984, −7.19898663168773333458136178286, −6.11894307726944117868007126211, −4.99115323643859005925276715641, −3.94915656690971544485075873478, −2.20235387345610512237403497567, −1.04660452143801062597081913184, 2.46193357292479286672163091286, 3.82706886633116460058721891858, 5.05319303153609988716094518202, 5.47367861109291995825164364446, 6.73290633230196779087479283770, 8.159841814204895543635195918086, 8.809418428143425233461210343722, 10.15254557818818448318351944546, 11.19635810014721826645038662350, 11.96539239385485365058424390533

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