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Properties

Label 2-600-120.77-c1-0-28
Degree $2$
Conductor $600$
Sign $0.991 + 0.132i$
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  + (−1.18 − 0.771i)2-s + (1.57 + 0.713i)3-s + (0.808 + 1.82i)4-s + (−1.31 − 2.06i)6-s + (−1.44 − 1.44i)7-s + (0.454 − 2.79i)8-s + (1.98 + 2.25i)9-s + 0.641·11-s + (−0.0287 + 3.46i)12-s + (2.03 + 2.03i)13-s + (0.597 + 2.83i)14-s + (−2.69 + 2.95i)16-s + (4.37 − 4.37i)17-s + (−0.612 − 4.19i)18-s + 4.93·19-s + ⋯
L(s)  = 1  + (−0.837 − 0.545i)2-s + (0.911 + 0.411i)3-s + (0.404 + 0.914i)4-s + (−0.538 − 0.842i)6-s + (−0.546 − 0.546i)7-s + (0.160 − 0.987i)8-s + (0.660 + 0.750i)9-s + 0.193·11-s + (−0.00829 + 0.999i)12-s + (0.563 + 0.563i)13-s + (0.159 + 0.756i)14-s + (−0.673 + 0.739i)16-s + (1.06 − 1.06i)17-s + (−0.144 − 0.989i)18-s + 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.991 + 0.132i$
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: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39177 - 0.0925334i\)
\(L(\frac12)\) \(\approx\) \(1.39177 - 0.0925334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.771i)T \)
3 \( 1 + (-1.57 - 0.713i)T \)
5 \( 1 \)
good7 \( 1 + (1.44 + 1.44i)T + 7iT^{2} \)
11 \( 1 - 0.641T + 11T^{2} \)
13 \( 1 + (-2.03 - 2.03i)T + 13iT^{2} \)
17 \( 1 + (-4.37 + 4.37i)T - 17iT^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 + (3.73 + 3.73i)T + 23iT^{2} \)
29 \( 1 - 9.84iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + (-3.44 + 3.44i)T - 37iT^{2} \)
41 \( 1 - 7.20iT - 41T^{2} \)
43 \( 1 + (-4.37 - 4.37i)T + 43iT^{2} \)
47 \( 1 + (4.08 - 4.08i)T - 47iT^{2} \)
53 \( 1 + (3.83 - 3.83i)T - 53iT^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 + 9.64iT - 61T^{2} \)
67 \( 1 + (-2.59 + 2.59i)T - 67iT^{2} \)
71 \( 1 + 16.7iT - 71T^{2} \)
73 \( 1 + (8.40 - 8.40i)T - 73iT^{2} \)
79 \( 1 + 5.31iT - 79T^{2} \)
83 \( 1 + (-2.20 + 2.20i)T - 83iT^{2} \)
89 \( 1 + 3.96T + 89T^{2} \)
97 \( 1 + (-1.11 - 1.11i)T + 97iT^{2} \)
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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

−10.39194846210635056255737851016, −9.634135343915593656675555479864, −9.199780747221243882980639031752, −8.112196182782276699529773809386, −7.44372002751364011566835682928, −6.48962970543992873679873077292, −4.70769592131576647752559553917, −3.55773444192026099149856318418, −2.87728092913176452420585728666, −1.28492801071173044615336512936, 1.21065128376013908766765251730, 2.60479938390886805178520511747, 3.80951423908319985685841383081, 5.66668718360146504380180765378, 6.23688525043274893304021811639, 7.43992080862039046879577494562, 8.067359461068310895793126400730, 8.774143973927661771169122202233, 9.812790123149707325108915746062, 10.07665287379988426275701244753

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