Properties

Label 2-1200-15.14-c2-0-16
Degree $2$
Conductor $1200$
Sign $-0.0760 - 0.997i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 2.77i)3-s − 5.85i·7-s + (−6.42 − 6.29i)9-s − 12.4i·11-s + 11.8i·13-s − 29.2·17-s + 3.19·19-s + (16.2 + 6.64i)21-s + 19.5·23-s + (24.7 − 10.7i)27-s + 30.3i·29-s − 3.57·31-s + (34.6 + 14.1i)33-s + 42.7i·37-s + (−32.9 − 13.4i)39-s + ⋯
L(s)  = 1  + (−0.377 + 0.925i)3-s − 0.836i·7-s + (−0.714 − 0.699i)9-s − 1.13i·11-s + 0.912i·13-s − 1.72·17-s + 0.168·19-s + (0.774 + 0.316i)21-s + 0.849·23-s + (0.917 − 0.396i)27-s + 1.04i·29-s − 0.115·31-s + (1.04 + 0.428i)33-s + 1.15i·37-s + (−0.844 − 0.344i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0760 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0760 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0760 - 0.997i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.0760 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.139871066\)
\(L(\frac12)\) \(\approx\) \(1.139871066\)
\(L(2)\) 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 \( 1 + (1.13 - 2.77i)T \)
5 \( 1 \)
good7 \( 1 + 5.85iT - 49T^{2} \)
11 \( 1 + 12.4iT - 121T^{2} \)
13 \( 1 - 11.8iT - 169T^{2} \)
17 \( 1 + 29.2T + 289T^{2} \)
19 \( 1 - 3.19T + 361T^{2} \)
23 \( 1 - 19.5T + 529T^{2} \)
29 \( 1 - 30.3iT - 841T^{2} \)
31 \( 1 + 3.57T + 961T^{2} \)
37 \( 1 - 42.7iT - 1.36e3T^{2} \)
41 \( 1 - 6.39iT - 1.68e3T^{2} \)
43 \( 1 - 62.6iT - 1.84e3T^{2} \)
47 \( 1 - 69.3T + 2.20e3T^{2} \)
53 \( 1 - 57.7T + 2.80e3T^{2} \)
59 \( 1 + 78.9iT - 3.48e3T^{2} \)
61 \( 1 - 68.5T + 3.72e3T^{2} \)
67 \( 1 - 90.5iT - 4.48e3T^{2} \)
71 \( 1 - 26.5iT - 5.04e3T^{2} \)
73 \( 1 - 40.0iT - 5.32e3T^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 + 9.36T + 6.88e3T^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 - 161. iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.805559140700682644531725737024, −8.941322170740559647418580796812, −8.481308463111359930784201406128, −7.04691403954840336908617628893, −6.49499036269605690524519991468, −5.43030406224390516277540367399, −4.49962265133929892508810111819, −3.84467761128851770982405048223, −2.75754283063273541295135437001, −0.970188433539684069330006500584, 0.43068518991225412591775949330, 2.02319626438183584861793511163, 2.61955850508013348475784256792, 4.24893541414998419895063403882, 5.30705055726995742362767014385, 5.94324193524043637410285234388, 7.00593909729700961231962321361, 7.46026508757393507182010821853, 8.603560355424695520136288451594, 9.091195254171283384185375006497

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