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Properties

Label 2-1200-5.4-c1-0-9
Degree $2$
Conductor $1200$
Sign $0.894 + 0.447i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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 − 4i·7-s − 9-s + 6i·13-s − 2i·17-s + 4·19-s + 4·21-s − 8i·23-s − i·27-s + 6·29-s − 6i·37-s − 6·39-s + 10·41-s − 4i·43-s − 8i·47-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 1.66i·13-s − 0.485i·17-s + 0.917·19-s + 0.872·21-s − 1.66i·23-s − 0.192i·27-s + 1.11·29-s − 0.986i·37-s − 0.960·39-s + 1.56·41-s − 0.609i·43-s − 1.16i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574700616\)
\(L(\frac12)\) \(\approx\) \(1.574700616\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.760732041280972527856347353013, −9.014339424275162161084751947198, −8.085502191399673023672854069241, −7.02120669528881926224263578818, −6.63787881418398969169268982138, −5.23244228999453151157744291844, −4.31653337696327592254722173010, −3.82848213857089380918864401632, −2.42501775851794867666809742734, −0.790843854266940326961438179187, 1.23136801108951504994366663802, 2.62126564951844982826337065403, 3.28303566753476648187416380439, 4.95347451618516257564491111813, 5.72723282946656731861682842109, 6.23112452798933265342365838147, 7.63218049132107445008936487280, 8.001554278993529444070278222770, 8.980319802170805215763440321635, 9.654292270207471035205419719909

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