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Properties

Label 2-300-3.2-c8-0-29
Degree $2$
Conductor $300$
Sign $0.917 + 0.398i$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (74.2 + 32.2i)3-s − 4.01e3·7-s + (4.47e3 + 4.79e3i)9-s + 1.13e4i·11-s + 1.46e4·13-s − 1.21e5i·17-s − 2.28e4·19-s + (−2.98e5 − 1.29e5i)21-s − 4.52e5i·23-s + (1.77e5 + 5.00e5i)27-s + 9.25e5i·29-s − 1.24e6·31-s + (−3.65e5 + 8.41e5i)33-s + 9.96e3·37-s + (1.08e6 + 4.73e5i)39-s + ⋯
L(s)  = 1  + (0.917 + 0.398i)3-s − 1.67·7-s + (0.682 + 0.730i)9-s + 0.774i·11-s + 0.513·13-s − 1.45i·17-s − 0.175·19-s + (−1.53 − 0.666i)21-s − 1.61i·23-s + (0.334 + 0.942i)27-s + 1.30i·29-s − 1.34·31-s + (−0.308 + 0.709i)33-s + 0.00531·37-s + (0.471 + 0.204i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 0.917 + 0.398i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.134004199\)
\(L(\frac12)\) \(\approx\) \(2.134004199\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-74.2 - 32.2i)T \)
5 \( 1 \)
good7 \( 1 + 4.01e3T + 5.76e6T^{2} \)
11 \( 1 - 1.13e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.46e4T + 8.15e8T^{2} \)
17 \( 1 + 1.21e5iT - 6.97e9T^{2} \)
19 \( 1 + 2.28e4T + 1.69e10T^{2} \)
23 \( 1 + 4.52e5iT - 7.83e10T^{2} \)
29 \( 1 - 9.25e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.24e6T + 8.52e11T^{2} \)
37 \( 1 - 9.96e3T + 3.51e12T^{2} \)
41 \( 1 + 3.24e6iT - 7.98e12T^{2} \)
43 \( 1 - 9.86e5T + 1.16e13T^{2} \)
47 \( 1 - 7.49e5iT - 2.38e13T^{2} \)
53 \( 1 - 3.98e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.17e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.04e7T + 1.91e14T^{2} \)
67 \( 1 - 1.35e7T + 4.06e14T^{2} \)
71 \( 1 - 1.35e7iT - 6.45e14T^{2} \)
73 \( 1 - 5.39e7T + 8.06e14T^{2} \)
79 \( 1 + 1.08e7T + 1.51e15T^{2} \)
83 \( 1 - 5.54e7iT - 2.25e15T^{2} \)
89 \( 1 + 8.17e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.22e8T + 7.83e15T^{2} \)
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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

−10.04370523682241419920100702042, −9.348000405710896324175463644561, −8.676963120138019829396423956283, −7.27438215815462387554682925168, −6.67172337549677685322710049417, −5.19426204166831009171256153122, −3.98368324837895283721522452043, −3.10118858321203666775624475280, −2.19141932517357200297552898303, −0.47999217742439732664105325879, 0.830529694975455498544416170370, 2.12202556643126511112814345253, 3.42912289666735600071414156557, 3.75956405755010902627760978238, 5.86293230236412867555792437287, 6.47893741417048757592094160110, 7.60981585730967334278057244251, 8.564285408247522101170657768381, 9.410165577411366670904177201329, 10.12549695952715005013628220992

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