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Properties

Label 2-1200-15.14-c2-0-61
Degree $2$
Conductor $1200$
Sign $-0.992 - 0.123i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.82 + i)3-s − 7i·7-s + (7.00 − 5.65i)9-s − 8.48i·11-s − 25i·13-s − 25.4·17-s − 7·19-s + (7 + 19.7i)21-s + 25.4·23-s + (−14.1 + 23.0i)27-s + 42.4i·29-s + 7·31-s + (8.48 + 24i)33-s − 2i·37-s + (25 + 70.7i)39-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)3-s − i·7-s + (0.777 − 0.628i)9-s − 0.771i·11-s − 1.92i·13-s − 1.49·17-s − 0.368·19-s + (0.333 + 0.942i)21-s + 1.10·23-s + (−0.523 + 0.851i)27-s + 1.46i·29-s + 0.225·31-s + (0.257 + 0.727i)33-s − 0.0540i·37-s + (0.641 + 1.81i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.992 - 0.123i$
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: \((2,\ 1200,\ (\ :1),\ -0.992 - 0.123i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4173875947\)
\(L(\frac12)\) \(\approx\) \(0.4173875947\)
\(L(2)\) 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 \( 1 + (2.82 - i)T \)
5 \( 1 \)
good7 \( 1 + 7iT - 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 + 25iT - 169T^{2} \)
17 \( 1 + 25.4T + 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 - 25.4T + 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 - 7T + 961T^{2} \)
37 \( 1 + 2iT - 1.36e3T^{2} \)
41 \( 1 + 8.48iT - 1.68e3T^{2} \)
43 \( 1 + 41iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 59.3T + 2.80e3T^{2} \)
59 \( 1 - 33.9iT - 3.48e3T^{2} \)
61 \( 1 + T + 3.72e3T^{2} \)
67 \( 1 - 17iT - 4.48e3T^{2} \)
71 \( 1 - 42.4iT - 5.04e3T^{2} \)
73 \( 1 + 70iT - 5.32e3T^{2} \)
79 \( 1 + 58T + 6.24e3T^{2} \)
83 \( 1 + 118.T + 6.88e3T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 - 49iT - 9.40e3T^{2} \)
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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

−9.181341652313444350595862525179, −8.352634141174536082813198224479, −7.27723343523833493330386068131, −6.64278964451902735253415329976, −5.65011818765130617805514609886, −4.93108165515198497351579105964, −3.94755524780245079122928896907, −2.99214501888741818460519181355, −1.09540097078579645140606667992, −0.15869151030598197081948981726, 1.68771663962091490547487821082, 2.43831101400054183109312140101, 4.36684393713173736132082266467, 4.71503810520521658471173896153, 5.97069480624930357531892700891, 6.60181790402161353176180922733, 7.21982102644315627328214950739, 8.437091446325042243470049496563, 9.243392408689948481495068316053, 9.868510523033349492411751989661

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