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Properties

Label 2-600-5.3-c2-0-15
Degree $2$
Conductor $600$
Sign $0.525 + 0.850i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
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  + (1.22 + 1.22i)3-s + (5.44 − 5.44i)7-s + 2.99i·9-s − 6.44·11-s + (−14.4 − 14.4i)13-s + (23.1 − 23.1i)17-s − 16.6i·19-s + 13.3·21-s + (6.65 + 6.65i)23-s + (−3.67 + 3.67i)27-s + 0.0454i·29-s + 4.49·31-s + (−7.89 − 7.89i)33-s + (−35.3 + 35.3i)37-s − 35.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.778 − 0.778i)7-s + 0.333i·9-s − 0.586·11-s + (−1.11 − 1.11i)13-s + (1.36 − 1.36i)17-s − 0.878i·19-s + 0.635·21-s + (0.289 + 0.289i)23-s + (−0.136 + 0.136i)27-s + 0.00156i·29-s + 0.144·31-s + (−0.239 − 0.239i)33-s + (−0.955 + 0.955i)37-s − 0.907i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.925567422\)
\(L(\frac12)\) \(\approx\) \(1.925567422\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-5.44 + 5.44i)T - 49iT^{2} \)
11 \( 1 + 6.44T + 121T^{2} \)
13 \( 1 + (14.4 + 14.4i)T + 169iT^{2} \)
17 \( 1 + (-23.1 + 23.1i)T - 289iT^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
23 \( 1 + (-6.65 - 6.65i)T + 529iT^{2} \)
29 \( 1 - 0.0454iT - 841T^{2} \)
31 \( 1 - 4.49T + 961T^{2} \)
37 \( 1 + (35.3 - 35.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 20.2T + 1.68e3T^{2} \)
43 \( 1 + (32.2 + 32.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-50.5 + 50.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-5.50 - 5.50i)T + 2.80e3iT^{2} \)
59 \( 1 + 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (-85.2 + 85.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 48.4T + 5.04e3T^{2} \)
73 \( 1 + (-21.9 - 21.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (94.9 + 94.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (-37 + 37i)T - 9.40e3iT^{2} \)
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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

−10.18767531286968643966907568350, −9.680512279560523567724158019451, −8.450512974077294338863501524683, −7.63792536546336786050091749233, −7.10671589831243265174638120665, −5.20972829156717111691192639855, −4.99176131211493128421701594453, −3.49850674645614249570349280038, −2.51128942930985888177676149284, −0.70283963971282750097145809492, 1.57768307882658398817181535380, 2.52469675522747206476936201636, 3.92417834099491537321120827732, 5.14662998741369115743340052170, 5.99802802620394000440673620203, 7.23208962093387218638124238322, 8.004770867100845338968510096434, 8.687214026732482010078401020522, 9.691321377809416723475802984439, 10.52500929554778207676723056861

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