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Properties

Label 2-600-600.533-c1-0-24
Degree $2$
Conductor $600$
Sign $0.837 - 0.546i$
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 − 0.642i)2-s + (−1.71 − 0.270i)3-s + (1.17 + 1.61i)4-s + (0.901 + 2.04i)5-s + (1.98 + 1.43i)6-s + (0.854 − 0.854i)7-s + (−0.442 − 2.79i)8-s + (2.85 + 0.927i)9-s + (0.178 − 3.15i)10-s + (−0.160 − 0.493i)11-s + (−1.57 − 3.08i)12-s + (−1.62 + 0.528i)14-s + (−0.987 − 3.74i)15-s + (−1.23 + 3.80i)16-s + (−3 − 3i)18-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (−0.987 − 0.156i)3-s + (0.587 + 0.809i)4-s + (0.403 + 0.915i)5-s + (0.809 + 0.587i)6-s + (0.323 − 0.323i)7-s + (−0.156 − 0.987i)8-s + (0.951 + 0.309i)9-s + (0.0563 − 0.998i)10-s + (−0.0482 − 0.148i)11-s + (−0.453 − 0.891i)12-s + (−0.434 + 0.141i)14-s + (−0.254 − 0.966i)15-s + (−0.309 + 0.951i)16-s + (−0.707 − 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705258 + 0.209824i\)
\(L(\frac12)\) \(\approx\) \(0.705258 + 0.209824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.642i)T \)
3 \( 1 + (1.71 + 0.270i)T \)
5 \( 1 + (-0.901 - 2.04i)T \)
good7 \( 1 + (-0.854 + 0.854i)T - 7iT^{2} \)
11 \( 1 + (0.160 + 0.493i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (-5.49 - 7.56i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 1.12i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-12.2 - 1.93i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-14.0 - 4.57i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.34 - 2.64i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (2.92 + 4.01i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.30 - 14.5i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.07 - 19.4i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.53293668378523364994629549065, −10.31901124978624439950658290526, −9.204359623133285025974550140446, −8.041663717618390860041918787498, −7.08108136694079481416880353262, −6.56775774432895739457936193686, −5.39727226842609015004119294910, −3.96891718484824365091967189397, −2.60869933929403322452586183553, −1.22363409278762459367105187808, 0.72930985610253214763454242491, 2.08214326211426217772941815198, 4.41083100599725379161358671654, 5.32344077672584366959108181947, 6.00298794143966272518609356143, 6.95063033458520690448669806145, 8.075152689703431241007086504514, 8.832123796862402048217325522741, 9.844280606768389364640382893099, 10.25693202534820761440308673938

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