Properties

Label 2-1200-3.2-c2-0-64
Degree $2$
Conductor $1200$
Sign $0.0972 + 0.995i$
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  + (0.291 + 2.98i)3-s + 4.46·7-s + (−8.82 + 1.74i)9-s − 17.8i·11-s + 11.0·13-s + 0.794i·17-s − 26.5·19-s + (1.30 + 13.3i)21-s − 14.9i·23-s + (−7.77 − 25.8i)27-s − 5.58i·29-s − 53.1·31-s + (53.3 − 5.21i)33-s − 51.7·37-s + (3.21 + 32.8i)39-s + ⋯
L(s)  = 1  + (0.0972 + 0.995i)3-s + 0.637·7-s + (−0.981 + 0.193i)9-s − 1.62i·11-s + 0.846·13-s + 0.0467i·17-s − 1.39·19-s + (0.0619 + 0.634i)21-s − 0.649i·23-s + (−0.287 − 0.957i)27-s − 0.192i·29-s − 1.71·31-s + (1.61 − 0.157i)33-s − 1.39·37-s + (0.0823 + 0.842i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.0972 + 0.995i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.0972 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.034720878\)
\(L(\frac12)\) \(\approx\) \(1.034720878\)
\(L(2)\) 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 \( 1 + (-0.291 - 2.98i)T \)
5 \( 1 \)
good7 \( 1 - 4.46T + 49T^{2} \)
11 \( 1 + 17.8iT - 121T^{2} \)
13 \( 1 - 11.0T + 169T^{2} \)
17 \( 1 - 0.794iT - 289T^{2} \)
19 \( 1 + 26.5T + 361T^{2} \)
23 \( 1 + 14.9iT - 529T^{2} \)
29 \( 1 + 5.58iT - 841T^{2} \)
31 \( 1 + 53.1T + 961T^{2} \)
37 \( 1 + 51.7T + 1.36e3T^{2} \)
41 \( 1 + 67.8iT - 1.68e3T^{2} \)
43 \( 1 + 40.8T + 1.84e3T^{2} \)
47 \( 1 + 12.3iT - 2.20e3T^{2} \)
53 \( 1 - 37.0iT - 2.80e3T^{2} \)
59 \( 1 + 61.0iT - 3.48e3T^{2} \)
61 \( 1 - 97.8T + 3.72e3T^{2} \)
67 \( 1 - 3.02T + 4.48e3T^{2} \)
71 \( 1 + 57.0iT - 5.04e3T^{2} \)
73 \( 1 - 31.4T + 5.32e3T^{2} \)
79 \( 1 - 2.16T + 6.24e3T^{2} \)
83 \( 1 - 13.0iT - 6.88e3T^{2} \)
89 \( 1 - 173. iT - 7.92e3T^{2} \)
97 \( 1 - 91.6T + 9.40e3T^{2} \)
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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

−9.182982939259388406161571649848, −8.533061233690061934271336880633, −8.205429766097006023379488569302, −6.72240439189961789962466152516, −5.80280169424138795366198717116, −5.13499873572905998888790188500, −3.97143373521287650025092893608, −3.39906540223322574100606659914, −2.02156572510541823525350677449, −0.28595368234134848833708724313, 1.51578314056836495300853496257, 2.09124442245925906429052609781, 3.53174872664939624530941930878, 4.66265062206701386572841883434, 5.59885648973217779999940773955, 6.65795829042883989865910533061, 7.20449843296804396913667938874, 8.105938775740010596724402941720, 8.713955850950206785270143056888, 9.664644405438894846596173236197

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