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Properties

Label 2-600-15.14-c2-0-6
Degree $2$
Conductor $600$
Sign $0.727 - 0.685i$
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  + (−0.864 − 2.87i)3-s − 9.02i·7-s + (−7.50 + 4.96i)9-s + 21.8i·11-s + 21.6i·13-s − 12.1·17-s − 3.03·19-s + (−25.9 + 7.80i)21-s + 28.5·23-s + (20.7 + 17.2i)27-s − 12.0i·29-s + 2.19·31-s + (62.8 − 18.9i)33-s − 0.839i·37-s + (62.2 − 18.7i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.957i)3-s − 1.28i·7-s + (−0.833 + 0.551i)9-s + 1.98i·11-s + 1.66i·13-s − 0.712·17-s − 0.159·19-s + (−1.23 + 0.371i)21-s + 1.24·23-s + (0.768 + 0.639i)27-s − 0.415i·29-s + 0.0706·31-s + (1.90 − 0.573i)33-s − 0.0227i·37-s + (1.59 − 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.727 - 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.105306340\)
\(L(\frac12)\) \(\approx\) \(1.105306340\)
\(L(2)\) 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 \( 1 + (0.864 + 2.87i)T \)
5 \( 1 \)
good7 \( 1 + 9.02iT - 49T^{2} \)
11 \( 1 - 21.8iT - 121T^{2} \)
13 \( 1 - 21.6iT - 169T^{2} \)
17 \( 1 + 12.1T + 289T^{2} \)
19 \( 1 + 3.03T + 361T^{2} \)
23 \( 1 - 28.5T + 529T^{2} \)
29 \( 1 + 12.0iT - 841T^{2} \)
31 \( 1 - 2.19T + 961T^{2} \)
37 \( 1 + 0.839iT - 1.36e3T^{2} \)
41 \( 1 - 35.5iT - 1.68e3T^{2} \)
43 \( 1 + 12.7iT - 1.84e3T^{2} \)
47 \( 1 - 22.5T + 2.20e3T^{2} \)
53 \( 1 + 9.13T + 2.80e3T^{2} \)
59 \( 1 - 80.4iT - 3.48e3T^{2} \)
61 \( 1 + 57.8T + 3.72e3T^{2} \)
67 \( 1 - 63.0iT - 4.48e3T^{2} \)
71 \( 1 - 17.0iT - 5.04e3T^{2} \)
73 \( 1 - 52.1iT - 5.32e3T^{2} \)
79 \( 1 - 7.46T + 6.24e3T^{2} \)
83 \( 1 - 82.3T + 6.88e3T^{2} \)
89 \( 1 - 27.5iT - 7.92e3T^{2} \)
97 \( 1 + 114. iT - 9.40e3T^{2} \)
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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

−10.67994914072398278704320012672, −9.718225551724340889593438438210, −8.811928963446800114105772914914, −7.45880033884677438016670324136, −7.09631584917705794456659626632, −6.43546821870595050206444816653, −4.81461608156571618526423102083, −4.18499096760544698910709261570, −2.33091621564285523759314927934, −1.30839344955684539250772742069, 0.45797623242366102475548766251, 2.80747027610136331118804821896, 3.43490624077475963300146453671, 5.03043250438093663088672585134, 5.64205380436047316176666125730, 6.34609911881246871625866826243, 8.070388633863814589238318275590, 8.763611585977963546445024577105, 9.299824713911468396267671130242, 10.68918730254839649944172697001

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