Properties

Label 2-1200-5.4-c1-0-16
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 + 2i·13-s − 6i·17-s − 4·19-s − 4·21-s + i·27-s + 6·29-s − 8·31-s − 2i·37-s + 2·39-s − 6·41-s + 4i·43-s − 9·49-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 0.554i·13-s − 1.45i·17-s − 0.917·19-s − 0.872·21-s + 0.192i·27-s + 1.11·29-s − 1.43·31-s − 0.328i·37-s + 0.320·39-s − 0.937·41-s + 0.609i·43-s − 1.28·49-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

\(L(1)\) \(\approx\) \(1.059979109\)
\(L(\frac12)\) \(\approx\) \(1.059979109\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
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

−9.412415368895010110577584049983, −8.513959693519003072099003833275, −7.53290580843691848524365423607, −7.03106501189624728289311285097, −6.31622233499071781882115027900, −5.00958422924308679007974735678, −4.19621709129075032138817413281, −3.11151715435156831813944606012, −1.75916814902390267633954093228, −0.43816537375513400236523009121, 1.90070444860112036439956286397, 2.95446901053280278700842005809, 4.00079934849080517680346014977, 5.09997914259797008869091954507, 5.82500079550649568890455986869, 6.52816372776241258618775144025, 7.923960150013263745472702459032, 8.634492796115289623641541591247, 9.102583885333548575319203067640, 10.19174573345121234503406816936

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