Properties

Label 2-600-5.4-c1-0-0
Degree $2$
Conductor $600$
Sign $-0.894 - 0.447i$
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  + i·3-s + 5i·7-s − 9-s − 6·11-s − 3i·13-s + 2i·17-s − 19-s − 5·21-s − 2i·23-s i·27-s − 6·29-s + 3·31-s − 6i·33-s + 6i·37-s + 3·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.88i·7-s − 0.333·9-s − 1.80·11-s − 0.832i·13-s + 0.485i·17-s − 0.229·19-s − 1.09·21-s − 0.417i·23-s − 0.192i·27-s − 1.11·29-s + 0.538·31-s − 1.04i·33-s + 0.986i·37-s + 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195881 + 0.829769i\)
\(L(\frac12)\) \(\approx\) \(0.195881 + 0.829769i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 - iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−10.95925970321933935036357348622, −10.15006075759147659366982183866, −9.330922705046043833938664371728, −8.345507522572148688284547905203, −7.895026211609946857742827648042, −6.17774902954367476665381116904, −5.50400654566528170230515282740, −4.77633521378316265655297466110, −3.08745892033247913453334961768, −2.37240909098392244819698094744, 0.44493886281827968706292317175, 2.10768280699253065414019768653, 3.55985029952620517947321345456, 4.63144159076330806147000252771, 5.72923764441056247193135998389, 7.16363970047527631201995177066, 7.31710579453034447422737106350, 8.305381023859202545485121613592, 9.537490635967879284215278258656, 10.51100617914757906596746223324

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