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Properties

Label 2-600-8.5-c1-0-34
Degree $2$
Conductor $600$
Sign $0.398 + 0.917i$
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  + (1.40 − 0.192i)2-s − i·3-s + (1.92 − 0.540i)4-s + (−0.192 − 1.40i)6-s − 0.0802·7-s + (2.59 − 1.12i)8-s − 9-s − 2.41i·11-s + (−0.540 − 1.92i)12-s − 5.26i·13-s + (−0.112 + 0.0154i)14-s + (3.41 − 2.08i)16-s + 0.255·17-s + (−1.40 + 0.192i)18-s + 6.95i·19-s + ⋯
L(s)  = 1  + (0.990 − 0.136i)2-s − 0.577i·3-s + (0.962 − 0.270i)4-s + (−0.0786 − 0.571i)6-s − 0.0303·7-s + (0.917 − 0.398i)8-s − 0.333·9-s − 0.728i·11-s + (−0.155 − 0.555i)12-s − 1.46i·13-s + (−0.0300 + 0.00413i)14-s + (0.854 − 0.520i)16-s + 0.0620·17-s + (−0.330 + 0.0454i)18-s + 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27549 - 1.49179i\)
\(L(\frac12)\) \(\approx\) \(2.27549 - 1.49179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.192i)T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + 0.0802T + 7T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 + 5.26iT - 13T^{2} \)
17 \( 1 - 0.255T + 17T^{2} \)
19 \( 1 - 6.95iT - 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 - 4.51iT - 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 + 2.67iT - 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 - 4.08iT - 43T^{2} \)
47 \( 1 + 5.70T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 7.27iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 - 9.20iT - 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 8.50T + 97T^{2} \)
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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

−10.61246431462883762880249981631, −10.01517045481790130226492739047, −8.400216843301244687257348875776, −7.80135800891284140385168471616, −6.67331267801090181114933307380, −5.85363123114412341122510244013, −5.10572835819636042771886965541, −3.63278537739635929587819341388, −2.82022599728342630765980397423, −1.26536032103428077990105296102, 2.06344209376764578955067274414, 3.27294101449075602214325604802, 4.56326286695023169271860919684, 4.85211888659545454257564644107, 6.36484031646291433246859473959, 6.90254922922900074734787552781, 8.076167021429808551446954007725, 9.207253999410703496590047843610, 10.01164042839795975037108095439, 11.10444777612684505350914447107

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