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Properties

Label 2-600-120.59-c1-0-17
Degree $2$
Conductor $600$
Sign $0.793 + 0.607i$
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 + (2.10 + 1.25i)6-s − 4.34·7-s + (1.40 + 2.45i)8-s + (−0.448 − 2.96i)9-s − 1.83i·11-s + (1.23 − 3.23i)12-s − 0.588·13-s + (1.06 + 6.05i)14-s + (3.07 − 2.55i)16-s + 5.37·17-s + (−4.02 + 1.34i)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 + (0.859 + 0.511i)6-s − 1.64·7-s + (0.497 + 0.867i)8-s + (−0.149 − 0.988i)9-s − 0.553i·11-s + (0.355 − 0.934i)12-s − 0.163·13-s + (0.283 + 1.61i)14-s + (0.768 − 0.639i)16-s + 1.30·17-s + (−0.948 + 0.317i)18-s + 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.793 + 0.607i$
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: \((2,\ 600,\ (\ :1/2),\ 0.793 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.717064 - 0.243006i\)
\(L(\frac12)\) \(\approx\) \(0.717064 - 0.243006i\)
\(L(\frac{3}{2})\) 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

−10.37179523292835887781788896105, −9.854990046732575313933447221897, −9.361315520580336275173249862868, −8.252892153284964058147461289401, −6.86051377076109037379126167763, −5.77860746354516697014082653439, −4.91893092484690078095479159834, −3.41947870233421056688286910598, −3.20534938799990152328001549870, −0.78723269495648563941305357841, 0.831944006478771925926566276970, 2.98632858681692653036587365944, 4.48987098007898936019008538300, 5.73180234304036293033520210516, 6.20192691664573754613796449989, 7.23092197623826593542444073472, 7.65788645733749013991374928593, 8.969011640271019729204095932465, 9.896627452536933067587363889981, 10.36448242269168859983512415903

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