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Properties

Label 2-600-3.2-c2-0-18
Degree $2$
Conductor $600$
Sign $0.107 + 0.994i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.98 + 0.323i)3-s − 4.72·7-s + (8.79 − 1.92i)9-s + 4.76i·11-s + 1.06·13-s + 26.7i·17-s − 8.12·19-s + (14.0 − 1.52i)21-s − 40.0i·23-s + (−25.5 + 8.59i)27-s − 20.8i·29-s − 33.7·31-s + (−1.53 − 14.2i)33-s + 60.4·37-s + (−3.18 + 0.344i)39-s + ⋯
L(s)  = 1  + (−0.994 + 0.107i)3-s − 0.675·7-s + (0.976 − 0.214i)9-s + 0.433i·11-s + 0.0820·13-s + 1.57i·17-s − 0.427·19-s + (0.671 − 0.0727i)21-s − 1.74i·23-s + (−0.948 + 0.318i)27-s − 0.719i·29-s − 1.08·31-s + (−0.0466 − 0.430i)33-s + 1.63·37-s + (−0.0815 + 0.00884i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.107 + 0.994i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.107 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6983020006\)
\(L(\frac12)\) \(\approx\) \(0.6983020006\)
\(L(2)\) 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 \( 1 + (2.98 - 0.323i)T \)
5 \( 1 \)
good7 \( 1 + 4.72T + 49T^{2} \)
11 \( 1 - 4.76iT - 121T^{2} \)
13 \( 1 - 1.06T + 169T^{2} \)
17 \( 1 - 26.7iT - 289T^{2} \)
19 \( 1 + 8.12T + 361T^{2} \)
23 \( 1 + 40.0iT - 529T^{2} \)
29 \( 1 + 20.8iT - 841T^{2} \)
31 \( 1 + 33.7T + 961T^{2} \)
37 \( 1 - 60.4T + 1.36e3T^{2} \)
41 \( 1 + 59.2iT - 1.68e3T^{2} \)
43 \( 1 - 56.4T + 1.84e3T^{2} \)
47 \( 1 + 9.68iT - 2.20e3T^{2} \)
53 \( 1 + 93.1iT - 2.80e3T^{2} \)
59 \( 1 + 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 57.7T + 3.72e3T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + 90.1iT - 5.04e3T^{2} \)
73 \( 1 + 40.0T + 5.32e3T^{2} \)
79 \( 1 - 65.3T + 6.24e3T^{2} \)
83 \( 1 + 117. iT - 6.88e3T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 - 15.2T + 9.40e3T^{2} \)
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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

−10.41809210314143977759951621379, −9.596820833214067124626389252360, −8.552943724643319356116608310829, −7.41167042772799994184312032188, −6.39403601035264613730163048271, −5.91459058776438547850744937864, −4.60240000724083418972411217702, −3.78422190757941672074571253237, −2.06396180912511721169161700622, −0.34905282372581582969855430857, 1.10294984421786921699854090700, 2.88042918642902451890229992215, 4.17215954476018252720981029266, 5.32173813830728738947583217423, 6.04730757490386904859905861716, 7.03198100019533134430152515777, 7.74083356680059211999451195583, 9.263672172686371129924020998916, 9.697715794549273802423358302384, 10.90984805933705184694657410489

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