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Properties

Label 2-600-15.8-c1-0-12
Degree $2$
Conductor $600$
Sign $0.775 + 0.631i$
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  + (1.47 − 0.912i)3-s + (1.78 − 1.78i)7-s + (1.33 − 2.68i)9-s + 4.25i·11-s + (2.90 + 2.90i)13-s + (0.443 + 0.443i)17-s − 7.74i·19-s + (1 − 4.25i)21-s + (1.94 − 1.94i)23-s + (−0.483 − 5.17i)27-s − 10.0·29-s + 0.372·31-s + (3.88 + 6.26i)33-s + (1.12 − 1.12i)37-s + (6.92 + 1.62i)39-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)3-s + (0.674 − 0.674i)7-s + (0.445 − 0.895i)9-s + 1.28i·11-s + (0.805 + 0.805i)13-s + (0.107 + 0.107i)17-s − 1.77i·19-s + (0.218 − 0.928i)21-s + (0.404 − 0.404i)23-s + (−0.0929 − 0.995i)27-s − 1.87·29-s + 0.0668·31-s + (0.675 + 1.09i)33-s + (0.184 − 0.184i)37-s + (1.10 + 0.260i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01123 - 0.715583i\)
\(L(\frac12)\) \(\approx\) \(2.01123 - 0.715583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.47 + 0.912i)T \)
5 \( 1 \)
good7 \( 1 + (-1.78 + 1.78i)T - 7iT^{2} \)
11 \( 1 - 4.25iT - 11T^{2} \)
13 \( 1 + (-2.90 - 2.90i)T + 13iT^{2} \)
17 \( 1 + (-0.443 - 0.443i)T + 17iT^{2} \)
19 \( 1 + 7.74iT - 19T^{2} \)
23 \( 1 + (-1.94 + 1.94i)T - 23iT^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 - 0.372T + 31T^{2} \)
37 \( 1 + (-1.12 + 1.12i)T - 37iT^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + (-6.68 - 6.68i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (7.59 - 7.59i)T - 53iT^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 4.37T + 61T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (7.90 + 7.90i)T + 73iT^{2} \)
79 \( 1 - 2.74iT - 79T^{2} \)
83 \( 1 + (-8.04 + 8.04i)T - 83iT^{2} \)
89 \( 1 - 0.497T + 89T^{2} \)
97 \( 1 + (6.47 - 6.47i)T - 97iT^{2} \)
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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

−10.67492118965800571318622374567, −9.286868714786808706367852329642, −9.053126008923231639676331880216, −7.64751613460032023987782843643, −7.31075494832463537977190124779, −6.30839092652892701086453643567, −4.69950078119880145938155089183, −3.96908026052169713161443550284, −2.48821100162687774360526016096, −1.35831102808621896134196587081, 1.68973182162214200329364860802, 3.13658311718407493953025403688, 3.87720302251507435839151207844, 5.35241436119959707399982549332, 5.91540291920098464987211032065, 7.62046044038080903771158870306, 8.252454805654337774776832784543, 8.857745512893801659280920103780, 9.808325433832048283395580425911, 10.79834758281475971935795350407

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