Properties

Label 2-1200-3.2-c2-0-53
Degree $2$
Conductor $1200$
Sign $0.971 + 0.235i$
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  + (2.91 + 0.707i)3-s + 5.83·7-s + (8 + 4.12i)9-s − 16.4i·11-s + 11.3i·17-s + 12·19-s + (17 + 4.12i)21-s − 24.0i·23-s + (20.4 + 17.6i)27-s + 32·31-s + (11.6 − 48.0i)33-s − 23.3·37-s − 57.7i·41-s + 40.8·43-s − 35.3i·47-s + ⋯
L(s)  = 1  + (0.971 + 0.235i)3-s + 0.832·7-s + (0.888 + 0.458i)9-s − 1.49i·11-s + 0.665i·17-s + 0.631·19-s + (0.809 + 0.196i)21-s − 1.04i·23-s + (0.755 + 0.654i)27-s + 1.03·31-s + (0.353 − 1.45i)33-s − 0.630·37-s − 1.40i·41-s + 0.949·43-s − 0.752i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.971 + 0.235i$
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: \((2,\ 1200,\ (\ :1),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.290183062\)
\(L(\frac12)\) \(\approx\) \(3.290183062\)
\(L(2)\) 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 \( 1 + (-2.91 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 - 5.83T + 49T^{2} \)
11 \( 1 + 16.4iT - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 11.3iT - 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 + 24.0iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 32T + 961T^{2} \)
37 \( 1 + 23.3T + 1.36e3T^{2} \)
41 \( 1 + 57.7iT - 1.68e3T^{2} \)
43 \( 1 - 40.8T + 1.84e3T^{2} \)
47 \( 1 + 35.3iT - 2.20e3T^{2} \)
53 \( 1 - 67.8iT - 2.80e3T^{2} \)
59 \( 1 - 16.4iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 + 5.83T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 116.T + 5.32e3T^{2} \)
79 \( 1 + 72T + 6.24e3T^{2} \)
83 \( 1 - 43.8iT - 6.88e3T^{2} \)
89 \( 1 - 65.9iT - 7.92e3T^{2} \)
97 \( 1 - 163.T + 9.40e3T^{2} \)
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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

−9.329401043249144140480194845840, −8.540312570601821420223726674985, −8.191143999667589106444268097133, −7.26804307180761866137214330275, −6.16109198759364986931253941388, −5.16765947784305652682658031117, −4.17917065910469650127690282118, −3.29980427480520679415704628265, −2.28563088645017281250323724353, −1.00292540316474624783634881548, 1.28808868642069214270100484425, 2.19290996970596016459492930421, 3.27034247340394124367875606633, 4.44690160496100794466031510392, 5.06350918940668859887237603715, 6.48581659268062826958060133545, 7.44808330435580597415645397282, 7.76861706763498426903012439311, 8.757962352551235345529043417405, 9.642116467732150666327059692543

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