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Properties

Label 2-600-24.11-c1-0-19
Degree $2$
Conductor $600$
Sign $-0.215 - 0.976i$
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  + (0.847 + 1.13i)2-s + (0.242 − 1.71i)3-s + (−0.562 + 1.91i)4-s + (2.14 − 1.17i)6-s + 3.08i·7-s + (−2.64 + 0.990i)8-s + (−2.88 − 0.831i)9-s + 2.54i·11-s + (3.15 + 1.42i)12-s + 5.06i·13-s + (−3.49 + 2.61i)14-s + (−3.36 − 2.15i)16-s + 0.214i·17-s + (−1.50 − 3.96i)18-s + 2.60·19-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)2-s + (0.139 − 0.990i)3-s + (−0.281 + 0.959i)4-s + (0.876 − 0.481i)6-s + 1.16i·7-s + (−0.936 + 0.350i)8-s + (−0.960 − 0.277i)9-s + 0.767i·11-s + (0.910 + 0.412i)12-s + 1.40i·13-s + (−0.934 + 0.700i)14-s + (−0.841 − 0.539i)16-s + 0.0519i·17-s + (−0.354 − 0.935i)18-s + 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11152 + 1.38379i\)
\(L(\frac12)\) \(\approx\) \(1.11152 + 1.38379i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.847 - 1.13i)T \)
3 \( 1 + (-0.242 + 1.71i)T \)
5 \( 1 \)
good7 \( 1 - 3.08iT - 7T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 - 5.06iT - 13T^{2} \)
17 \( 1 - 0.214iT - 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 + 4.58iT - 31T^{2} \)
37 \( 1 - 7.67iT - 37T^{2} \)
41 \( 1 + 9.26iT - 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 9.51T + 53T^{2} \)
59 \( 1 + 0.428iT - 59T^{2} \)
61 \( 1 + 1.11iT - 61T^{2} \)
67 \( 1 + 2.35T + 67T^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 - 2.29iT - 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 - 8.04T + 97T^{2} \)
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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

−11.54843384147361643558712445565, −9.677568068153571369602785878584, −8.813589030019283005440623625212, −8.218370852877388933508177848419, −7.03057792035590410471169935517, −6.61436616794968128809555287804, −5.55283187041387901697565234190, −4.63963973223851672048958524856, −3.11688661049304838525538616355, −2.02384253399343085356714966890, 0.826948284132675442116671838772, 2.99878289579363774861597315236, 3.51541310639866790298603520559, 4.71271460289165907571114210901, 5.38558690967334919599321596926, 6.55981126399915134540379070510, 7.977007827383717410603024315649, 8.906406199531293944113967705056, 10.00361274557109078054972507910, 10.43457552675300536983187063654

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