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Properties

Label 2-600-24.11-c1-0-33
Degree $2$
Conductor $600$
Sign $-0.269 + 0.963i$
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  + (−1.26 − 0.639i)2-s + (−1.57 − 0.730i)3-s + (1.18 + 1.61i)4-s + (1.51 + 1.92i)6-s − 1.25i·7-s + (−0.458 − 2.79i)8-s + (1.93 + 2.29i)9-s − 3.02i·11-s + (−0.677 − 3.39i)12-s + 5.65i·13-s + (−0.803 + 1.58i)14-s + (−1.20 + 3.81i)16-s − 2.45i·17-s + (−0.972 − 4.12i)18-s + 1.77·19-s + ⋯
L(s)  = 1  + (−0.891 − 0.452i)2-s + (−0.906 − 0.421i)3-s + (0.590 + 0.806i)4-s + (0.618 + 0.786i)6-s − 0.474i·7-s + (−0.162 − 0.986i)8-s + (0.644 + 0.764i)9-s − 0.911i·11-s + (−0.195 − 0.980i)12-s + 1.56i·13-s + (−0.214 + 0.423i)14-s + (−0.301 + 0.953i)16-s − 0.595i·17-s + (−0.229 − 0.973i)18-s + 0.407·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.269 + 0.963i$
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: \((2,\ 600,\ (\ :1/2),\ -0.269 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380721 - 0.501632i\)
\(L(\frac12)\) \(\approx\) \(0.380721 - 0.501632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.639i)T \)
3 \( 1 + (1.57 + 0.730i)T \)
5 \( 1 \)
good7 \( 1 + 1.25iT - 7T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 - 5.65iT - 13T^{2} \)
17 \( 1 + 2.45iT - 17T^{2} \)
19 \( 1 - 1.77T + 19T^{2} \)
23 \( 1 - 8.84T + 23T^{2} \)
29 \( 1 + 3.79T + 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + 6.45iT - 37T^{2} \)
41 \( 1 + 7.57iT - 41T^{2} \)
43 \( 1 + 4.37T + 43T^{2} \)
47 \( 1 + 1.83T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 4.91iT - 59T^{2} \)
61 \( 1 + 8.16iT - 61T^{2} \)
67 \( 1 - 8.50T + 67T^{2} \)
71 \( 1 + 7.00T + 71T^{2} \)
73 \( 1 - 4.59T + 73T^{2} \)
79 \( 1 + 7.36iT - 79T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−10.63709825238458490263309353713, −9.513436044381979900490121932769, −8.870843232277833494832013472422, −7.57994320503719563759086308591, −7.02825556339472266308137612231, −6.14055764121911504793782028227, −4.80427642383966358248829222947, −3.52847304599005339497809960231, −1.95557638793028373522734400998, −0.61207054050058817374141462889, 1.23214740299701940463851347329, 3.05440291993346435052571984818, 4.88318042156133675824658420748, 5.47307603093606368057021940033, 6.48448712446389912056992535937, 7.32126164367695709332229490990, 8.320318891148593242006858075217, 9.323068053920833662671014792877, 10.04788720263760360196793353133, 10.71543950107459258953286886784

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