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Properties

Label 2-600-15.8-c1-0-16
Degree $2$
Conductor $600$
Sign $-0.410 + 0.911i$
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  + (0.912 − 1.47i)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 + (−5.17 − 0.483i)27-s + 10.0·29-s + 0.372·31-s + (−6.26 − 3.88i)33-s + (−1.12 + 1.12i)37-s + (−6.92 + 1.62i)39-s + ⋯
L(s)  = 1  + (0.526 − 0.850i)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 + (−0.995 − 0.0929i)27-s + 1.87·29-s + 0.0668·31-s + (−1.09 − 0.675i)33-s + (−0.184 + 0.184i)37-s + (−1.10 + 0.260i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.410 + 0.911i$
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: \((2,\ 600,\ (\ :1/2),\ -0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726765 - 1.12479i\)
\(L(\frac12)\) \(\approx\) \(0.726765 - 1.12479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.912 + 1.47i)T \)
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

−10.35765784238296145817007484645, −9.230946194244451674797055926148, −8.665774889590017557595148103193, −7.81035707884281111541645045760, −6.72615235371420964368318511551, −6.07807681535267593433694856905, −4.90693113633829257139823776105, −3.09289817818217856416126331871, −2.69635985676657727822601841479, −0.68927071440391985771427048553, 2.03712264665801539213149553039, 3.40026438937178725862024377366, 4.29630167949343534169081534169, 5.15079188190792825835470118219, 6.59247104677429476151941249636, 7.42778372074816383908549292536, 8.368703884196835564729242891396, 9.529002576574929035187360195289, 9.954484632626761054927091127128, 10.54427727381098436137448502921

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