Properties

Label 2-600-120.59-c1-0-22
Degree $2$
Conductor $600$
Sign $0.123 + 0.992i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.244 − 1.39i)2-s + (−1.12 − 1.31i)3-s + (−1.88 + 0.680i)4-s + (−1.55 + 1.89i)6-s + 4.34·7-s + (1.40 + 2.45i)8-s + (−0.448 + 2.96i)9-s + 1.83i·11-s + (3.01 + 1.70i)12-s + 0.588·13-s + (−1.06 − 6.05i)14-s + (3.07 − 2.55i)16-s + 5.37·17-s + (4.24 − 0.0995i)18-s + 5.38·19-s + ⋯
L(s)  = 1  + (−0.172 − 0.984i)2-s + (−0.652 − 0.758i)3-s + (−0.940 + 0.340i)4-s + (−0.634 + 0.773i)6-s + 1.64·7-s + (0.497 + 0.867i)8-s + (−0.149 + 0.988i)9-s + 0.553i·11-s + (0.871 + 0.491i)12-s + 0.163·13-s + (−0.283 − 1.61i)14-s + (0.768 − 0.639i)16-s + 1.30·17-s + (0.999 − 0.0234i)18-s + 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.123 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915163 - 0.808374i\)
\(L(\frac12)\) \(\approx\) \(0.915163 - 0.808374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.244 + 1.39i)T \)
3 \( 1 + (1.12 + 1.31i)T \)
5 \( 1 \)
good7 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 - 1.83iT - 11T^{2} \)
13 \( 1 - 0.588T + 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 - 2.40iT - 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 - 7.06iT - 31T^{2} \)
37 \( 1 + 2.72T + 37T^{2} \)
41 \( 1 + 3.42iT - 41T^{2} \)
43 \( 1 + 2.96iT - 43T^{2} \)
47 \( 1 + 9.81iT - 47T^{2} \)
53 \( 1 + 6.65iT - 53T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + 9.27iT - 61T^{2} \)
67 \( 1 - 4.13iT - 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 4.42iT - 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 4.21iT - 89T^{2} \)
97 \( 1 - 2.16iT - 97T^{2} \)
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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

−10.70921643782168286556294323120, −9.853948146977565292345637493570, −8.663550760075738781843091907497, −7.79576280577587061347584643338, −7.25791476080308482246164870004, −5.32961884387744720898891784098, −5.16668383002884028863483272617, −3.64609429688795597208614307549, −2.01303977684601929125219846420, −1.19754611128187070779830194252, 1.09834592811992850397727788583, 3.61075026166575492178856929057, 4.68716452025964054915268482478, 5.41460331889776593282423015703, 6.06755587161415683154168051192, 7.51084171215998633903260640357, 8.047828615997959066231974914253, 9.120031624106442438750532799562, 9.839960806748055567291444290710, 10.88336901959208653782107763514

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