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Properties

Label 2-600-120.59-c1-0-39
Degree $2$
Conductor $600$
Sign $-0.338 + 0.940i$
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  + (−1.15 − 0.814i)2-s + (−0.887 + 1.48i)3-s + (0.672 + 1.88i)4-s + (2.23 − 0.995i)6-s − 0.797·7-s + (0.757 − 2.72i)8-s + (−1.42 − 2.64i)9-s + 0.320i·11-s + (−3.39 − 0.672i)12-s − 4.30·13-s + (0.921 + 0.649i)14-s + (−3.09 + 2.53i)16-s − 2.57·17-s + (−0.506 + 4.21i)18-s + 6.10·19-s + ⋯
L(s)  = 1  + (−0.817 − 0.576i)2-s + (−0.512 + 0.858i)3-s + (0.336 + 0.941i)4-s + (0.913 − 0.406i)6-s − 0.301·7-s + (0.267 − 0.963i)8-s + (−0.474 − 0.880i)9-s + 0.0966i·11-s + (−0.980 − 0.194i)12-s − 1.19·13-s + (0.246 + 0.173i)14-s + (−0.773 + 0.633i)16-s − 0.624·17-s + (−0.119 + 0.992i)18-s + 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.338 + 0.940i$
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: \((2,\ 600,\ (\ :1/2),\ -0.338 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205475 - 0.292386i\)
\(L(\frac12)\) \(\approx\) \(0.205475 - 0.292386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.814i)T \)
3 \( 1 + (0.887 - 1.48i)T \)
5 \( 1 \)
good7 \( 1 + 0.797T + 7T^{2} \)
11 \( 1 - 0.320iT - 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 - 6.10T + 19T^{2} \)
23 \( 1 + 3.13iT - 23T^{2} \)
29 \( 1 + 8.79T + 29T^{2} \)
31 \( 1 + 9.90iT - 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
41 \( 1 + 5.28iT - 41T^{2} \)
43 \( 1 + 2.97iT - 43T^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 + 3.94iT - 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 + 8.83iT - 61T^{2} \)
67 \( 1 + 4.66iT - 67T^{2} \)
71 \( 1 + 3.43T + 71T^{2} \)
73 \( 1 + 1.43iT - 73T^{2} \)
79 \( 1 + 2.89iT - 79T^{2} \)
83 \( 1 - 3.37T + 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 - 4.26iT - 97T^{2} \)
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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

−10.25057464653817532958093038990, −9.583577445019589727338625877599, −9.133015115993764433439543580743, −7.83425559610952237542704130827, −6.99523502659629355545648947554, −5.79871651533202336694439153708, −4.60013468987010833083938957142, −3.58546992791424496831620810461, −2.37895822866885816830078079434, −0.28470287858164367942346339029, 1.37745482804058425512077373682, 2.76645854160017676116640780899, 4.89720078435324326565055646684, 5.68005423374722025654642977765, 6.67357937941185633441676474874, 7.39234851214397817988541120990, 8.004675120810190768409077971076, 9.246341737509598550278885357068, 9.837495337723940259548365165164, 11.00284816801460561464905683041

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