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Properties

Label 2-600-120.59-c1-0-10
Degree $2$
Conductor $600$
Sign $-0.719 + 0.694i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.719 + 0.694i$
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: \((2,\ 600,\ (\ :1/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324168 - 0.803135i\)
\(L(\frac12)\) \(\approx\) \(0.324168 - 0.803135i\)
\(L(\frac{3}{2})\) 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

−10.95373239060792504783199505855, −9.829645397214021368191884438947, −9.429812706854661601666756682915, −8.733471997404842172597290851386, −7.53259314688749194189059432223, −6.82578737732303704907431278234, −5.78688264070379128928517512441, −4.70018853196518995314513974940, −3.74635271517275373037095912413, −2.80580829721764309511494216870, 0.40288577116316218674307388993, 2.15012174864485480456496167616, 3.18831069929541662818240646399, 3.86565871685890680202242762820, 5.57161255457674530758429687596, 6.46876229031594531072437493064, 7.49229260737399388560953773886, 8.669663113604534549681652767563, 9.432101733574673179278803140112, 9.864730407091846532057153551097

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