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Properties

Label 2-300-60.23-c2-0-20
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

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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: $\chi_{300} (143, \cdot )$
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 \( 1 + (-24.6 + 24.6i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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