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Properties

Label 2-1200-15.14-c2-0-36
Degree $2$
Conductor $1200$
Sign $0.451 + 0.892i$
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  + (−2.40 − 1.78i)3-s − 12.2i·7-s + (2.60 + 8.61i)9-s + 8.79i·11-s + 6.20i·13-s + 16.6·17-s − 8.93·19-s + (−21.8 + 29.3i)21-s + 40.1·23-s + (9.14 − 25.4i)27-s + 4.94i·29-s + 50.6·31-s + (15.7 − 21.1i)33-s + 41.3i·37-s + (11.0 − 14.9i)39-s + ⋯
L(s)  = 1  + (−0.802 − 0.596i)3-s − 1.74i·7-s + (0.288 + 0.957i)9-s + 0.799i·11-s + 0.477i·13-s + 0.977·17-s − 0.470·19-s + (−1.03 + 1.39i)21-s + 1.74·23-s + (0.338 − 0.940i)27-s + 0.170i·29-s + 1.63·31-s + (0.476 − 0.641i)33-s + 1.11i·37-s + (0.284 − 0.382i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.451 + 0.892i$
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: \((2,\ 1200,\ (\ :1),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.475858922\)
\(L(\frac12)\) \(\approx\) \(1.475858922\)
\(L(2)\) 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 \( 1 + (2.40 + 1.78i)T \)
5 \( 1 \)
good7 \( 1 + 12.2iT - 49T^{2} \)
11 \( 1 - 8.79iT - 121T^{2} \)
13 \( 1 - 6.20iT - 169T^{2} \)
17 \( 1 - 16.6T + 289T^{2} \)
19 \( 1 + 8.93T + 361T^{2} \)
23 \( 1 - 40.1T + 529T^{2} \)
29 \( 1 - 4.94iT - 841T^{2} \)
31 \( 1 - 50.6T + 961T^{2} \)
37 \( 1 - 41.3iT - 1.36e3T^{2} \)
41 \( 1 - 10.8iT - 1.68e3T^{2} \)
43 \( 1 - 39.5iT - 1.84e3T^{2} \)
47 \( 1 + 78.2T + 2.20e3T^{2} \)
53 \( 1 - 69.7T + 2.80e3T^{2} \)
59 \( 1 + 96.0iT - 3.48e3T^{2} \)
61 \( 1 - 26.7T + 3.72e3T^{2} \)
67 \( 1 + 66.5iT - 4.48e3T^{2} \)
71 \( 1 - 34.7iT - 5.04e3T^{2} \)
73 \( 1 + 70.5iT - 5.32e3T^{2} \)
79 \( 1 - 31.2T + 6.24e3T^{2} \)
83 \( 1 + 71.5T + 6.88e3T^{2} \)
89 \( 1 + 30.4iT - 7.92e3T^{2} \)
97 \( 1 + 108. iT - 9.40e3T^{2} \)
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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

−9.814342998253773078060490367466, −8.328112099666792280226723244539, −7.55612003788680758931179936485, −6.88744208316595870658932619262, −6.37761910928594419448218316021, −4.92431477035883142075701112846, −4.51119142765930025717235777347, −3.19664442834066375368669192350, −1.58337627090172641696944557789, −0.72217219690324084355948076517, 0.866654411415604036710262779516, 2.59907550272556817503506482277, 3.47271787800748895350762657164, 4.81741966328629129973055721345, 5.57568571623169232547907776389, 5.99538017295453777003434090372, 7.04758572246815421997211548330, 8.405276095400024971205463432850, 8.849810182169473533583185631895, 9.735528957348342941053090880654

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