Properties

Label 2-1200-5.3-c2-0-26
Degree $2$
Conductor $1200$
Sign $-0.229 + 0.973i$
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  + (−1.22 − 1.22i)3-s + (3.44 − 3.44i)7-s + 2.99i·9-s − 11.3·11-s + (5.55 + 5.55i)13-s + (17.3 − 17.3i)17-s + 8.69i·19-s − 8.44·21-s + (11.5 + 11.5i)23-s + (3.67 − 3.67i)27-s − 35.1i·29-s − 10.6·31-s + (13.8 + 13.8i)33-s + (6.04 − 6.04i)37-s − 13.5i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.492 − 0.492i)7-s + 0.333i·9-s − 1.03·11-s + (0.426 + 0.426i)13-s + (1.02 − 1.02i)17-s + 0.457i·19-s − 0.402·21-s + (0.502 + 0.502i)23-s + (0.136 − 0.136i)27-s − 1.21i·29-s − 0.345·31-s + (0.421 + 0.421i)33-s + (0.163 − 0.163i)37-s − 0.348i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.370120712\)
\(L(\frac12)\) \(\approx\) \(1.370120712\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-3.44 + 3.44i)T - 49iT^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (-5.55 - 5.55i)T + 169iT^{2} \)
17 \( 1 + (-17.3 + 17.3i)T - 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (-11.5 - 11.5i)T + 529iT^{2} \)
29 \( 1 + 35.1iT - 841T^{2} \)
31 \( 1 + 10.6T + 961T^{2} \)
37 \( 1 + (-6.04 + 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (26.4 + 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.696 - 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 + 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (45.1 - 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (77.7 + 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (-24.5 + 24.5i)T - 9.40e3iT^{2} \)
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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

−9.411188119362079391914394124154, −8.286165339867734787623327785880, −7.61375843101069720157653607483, −7.01198832574215205176709694162, −5.82050983149235629695174848480, −5.20420649178663143973088203296, −4.17767079679655014117329959557, −2.96520432291598868154298734446, −1.70338471001732256700438754124, −0.47876994088469545061825708378, 1.18184840826946378040987695054, 2.63578999032072642175749041473, 3.65030115971014969694633377534, 4.87627907550082784747357542811, 5.44745642748469215192040241208, 6.26468829056125915631962076391, 7.41830124262724111306664482224, 8.243651456035971954621225459290, 8.895803378622973596368560845501, 9.945582925373455454572558923464

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