Properties

Label 2-1200-5.4-c1-0-3
Degree $2$
Conductor $1200$
Sign $0.447 - 0.894i$
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 i·7-s − 9-s − 6·11-s + 5i·13-s + 6i·17-s + 5·19-s − 21-s + 6i·23-s + i·27-s + 6·29-s + 31-s + 6i·33-s − 2i·37-s + 5·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 1.80·11-s + 1.38i·13-s + 1.45i·17-s + 1.14·19-s − 0.218·21-s + 1.25i·23-s + 0.192i·27-s + 1.11·29-s + 0.179·31-s + 1.04i·33-s − 0.328i·37-s + 0.800·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.085277432\)
\(L(\frac12)\) \(\approx\) \(1.085277432\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
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

−9.959717629901749504190177564788, −9.026110966715894208987583814102, −8.027814440861803343804966104436, −7.56017366500586457688626847357, −6.64003420502952692194388130617, −5.72557191323100581696881148215, −4.84624624232527391956809797003, −3.69177777094725348577589227745, −2.54407075065313153693300716550, −1.39129501132030225555070081083, 0.47598048900205289374967925284, 2.72711960451348385144025008890, 3.01295475985222491424249264213, 4.72373992798063128931570292863, 5.19477180863916879687016083153, 6.01183768748722130748896607551, 7.36044079011193432919257285284, 7.967030258758887959038005237918, 8.794207683349364235624498809072, 9.742529921245462153067943482561

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