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Properties

Label 2-600-120.77-c1-0-64
Degree $2$
Conductor $600$
Sign $-0.229 - 0.973i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s − 1.73i·3-s + (1.73 + i)4-s + (−0.633 + 2.36i)6-s + (−3 − 3i)7-s + (−1.99 − 2i)8-s − 2.99·9-s − 3.46·11-s + (1.73 − 2.99i)12-s + (3.46 + 3.46i)13-s + (3 + 5.19i)14-s + (1.99 + 3.46i)16-s + (−4 + 4i)17-s + (4.09 + 1.09i)18-s + 3.46·19-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s − 0.999i·3-s + (0.866 + 0.5i)4-s + (−0.258 + 0.965i)6-s + (−1.13 − 1.13i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s − 1.04·11-s + (0.499 − 0.866i)12-s + (0.960 + 0.960i)13-s + (0.801 + 1.38i)14-s + (0.499 + 0.866i)16-s + (−0.970 + 0.970i)17-s + (0.965 + 0.258i)18-s + 0.794·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 \)
good7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + (-3.46 - 3.46i)T + 13iT^{2} \)
17 \( 1 + (4 - 4i)T - 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (1.73 + 1.73i)T + 43iT^{2} \)
47 \( 1 + (5 - 5i)T - 47iT^{2} \)
53 \( 1 + (-3.46 + 3.46i)T - 53iT^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + (5.19 - 5.19i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + (1.73 - 1.73i)T - 83iT^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (6 + 6i)T + 97iT^{2} \)
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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

−10.08495586350299035276181556065, −9.113798982925386597462129620732, −8.239937498225070216462844688811, −7.40545686288682513669883587165, −6.66667038379220174521711421834, −6.01243755803567177580865668824, −3.95665241319280244575890575718, −2.85419609020867071693304274718, −1.50329024133171333961405448836, 0, 2.57993781644300526256428154779, 3.31944107786767652339233543733, 5.28825644087335134144193804014, 5.71859329611369152266052286705, 6.83991228313777432514046497393, 8.086455352724038538696910517861, 8.881633204442117913421413066311, 9.453973672574801864273752054145, 10.23920082726794126833505641644

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