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Properties

Label 2-600-120.53-c1-0-18
Degree $2$
Conductor $600$
Sign $-0.834 - 0.550i$
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  + (1.30 + 0.552i)2-s + (−0.504 + 1.65i)3-s + (1.38 + 1.43i)4-s + (−1.57 + 1.87i)6-s + (−2.83 + 2.83i)7-s + (1.01 + 2.64i)8-s + (−2.48 − 1.67i)9-s + 4.74·11-s + (−3.08 + 1.57i)12-s + (0.867 − 0.867i)13-s + (−5.26 + 2.12i)14-s + (−0.136 + 3.99i)16-s + (−1.73 − 1.73i)17-s + (−2.31 − 3.55i)18-s − 3.35·19-s + ⋯
L(s)  = 1  + (0.920 + 0.390i)2-s + (−0.291 + 0.956i)3-s + (0.694 + 0.719i)4-s + (−0.641 + 0.766i)6-s + (−1.07 + 1.07i)7-s + (0.358 + 0.933i)8-s + (−0.829 − 0.557i)9-s + 1.43·11-s + (−0.890 + 0.455i)12-s + (0.240 − 0.240i)13-s + (−1.40 + 0.568i)14-s + (−0.0341 + 0.999i)16-s + (−0.419 − 0.419i)17-s + (−0.546 − 0.837i)18-s − 0.769·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.834 - 0.550i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572433 + 1.90770i\)
\(L(\frac12)\) \(\approx\) \(0.572433 + 1.90770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.552i)T \)
3 \( 1 + (0.504 - 1.65i)T \)
5 \( 1 \)
good7 \( 1 + (2.83 - 2.83i)T - 7iT^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 + (-0.867 + 0.867i)T - 13iT^{2} \)
17 \( 1 + (1.73 + 1.73i)T + 17iT^{2} \)
19 \( 1 + 3.35T + 19T^{2} \)
23 \( 1 + (4.82 - 4.82i)T - 23iT^{2} \)
29 \( 1 - 0.936iT - 29T^{2} \)
31 \( 1 - 5.49T + 31T^{2} \)
37 \( 1 + (0.749 + 0.749i)T + 37iT^{2} \)
41 \( 1 + 2.24iT - 41T^{2} \)
43 \( 1 + (-8.75 + 8.75i)T - 43iT^{2} \)
47 \( 1 + (-6.71 - 6.71i)T + 47iT^{2} \)
53 \( 1 + (-8.46 - 8.46i)T + 53iT^{2} \)
59 \( 1 - 6.53iT - 59T^{2} \)
61 \( 1 + 5.10iT - 61T^{2} \)
67 \( 1 + (-4.85 - 4.85i)T + 67iT^{2} \)
71 \( 1 - 8.34iT - 71T^{2} \)
73 \( 1 + (1.57 + 1.57i)T + 73iT^{2} \)
79 \( 1 + 7.75iT - 79T^{2} \)
83 \( 1 + (9.15 + 9.15i)T + 83iT^{2} \)
89 \( 1 - 4.97T + 89T^{2} \)
97 \( 1 + (-1.42 + 1.42i)T - 97iT^{2} \)
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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

−11.23985291492064704323701415151, −10.17588198996282265045509873014, −9.145214279177633040188934927083, −8.665679734113710542121510802821, −7.10248746199361475348415529004, −6.03920672110492637656970767586, −5.79539280743358464686765457582, −4.37601340907275830419790010526, −3.65272839831672507374485329666, −2.54953161627132461706589680243, 0.865645204341997432953199738710, 2.26384644413993856878661425588, 3.68856956031679084600285109091, 4.42554935302429713230373996265, 6.13225835137987595078669514208, 6.45203603652322083070952925322, 7.15221030030385663088679063327, 8.470962359496414085019467640464, 9.719761855923462619865027459516, 10.55769073572740399530630290217

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