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Properties

Label 2-1200-5.2-c2-0-2
Degree $2$
Conductor $1200$
Sign $-0.991 - 0.130i$
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 + (−0.775 − 0.775i)7-s − 2.99i·9-s − 2.89·11-s + (5.87 − 5.87i)13-s + (4.44 + 4.44i)17-s − 0.101i·19-s + 1.89·21-s + (−25.3 + 25.3i)23-s + (3.67 + 3.67i)27-s − 32.2i·29-s + 3.69·31-s + (3.55 − 3.55i)33-s + (42.6 + 42.6i)37-s + 14.3i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.110 − 0.110i)7-s − 0.333i·9-s − 0.263·11-s + (0.452 − 0.452i)13-s + (0.261 + 0.261i)17-s − 0.00531i·19-s + 0.0904·21-s + (−1.10 + 1.10i)23-s + (0.136 + 0.136i)27-s − 1.11i·29-s + 0.119·31-s + (0.107 − 0.107i)33-s + (1.15 + 1.15i)37-s + 0.369i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.991 - 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3269783480\)
\(L(\frac12)\) \(\approx\) \(0.3269783480\)
\(L(2)\) 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 \( 1 + (0.775 + 0.775i)T + 49iT^{2} \)
11 \( 1 + 2.89T + 121T^{2} \)
13 \( 1 + (-5.87 + 5.87i)T - 169iT^{2} \)
17 \( 1 + (-4.44 - 4.44i)T + 289iT^{2} \)
19 \( 1 + 0.101iT - 361T^{2} \)
23 \( 1 + (25.3 - 25.3i)T - 529iT^{2} \)
29 \( 1 + 32.2iT - 841T^{2} \)
31 \( 1 - 3.69T + 961T^{2} \)
37 \( 1 + (-42.6 - 42.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 12.8T + 1.68e3T^{2} \)
43 \( 1 + (49.2 - 49.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (2.85 + 2.85i)T + 2.20e3iT^{2} \)
53 \( 1 + (13.1 - 13.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 76.3iT - 3.48e3T^{2} \)
61 \( 1 + 103.T + 3.72e3T^{2} \)
67 \( 1 + (47.6 + 47.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 29.7T + 5.04e3T^{2} \)
73 \( 1 + (-3.50 + 3.50i)T - 5.32e3iT^{2} \)
79 \( 1 + 87.7iT - 6.24e3T^{2} \)
83 \( 1 + (81.7 - 81.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 96.5iT - 7.92e3T^{2} \)
97 \( 1 + (-54.2 - 54.2i)T + 9.40e3iT^{2} \)
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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

−10.02874523748906023169657968367, −9.317079475996592892680230192552, −8.197139313602988835767355521492, −7.65031034221632719428924140866, −6.37174733711220848196595566408, −5.83701680279340757981516828622, −4.82130953414174271897361389793, −3.89610533643209519376938058670, −2.92832272943218759079811889149, −1.40969149857387187415047990993, 0.10396537190795019219888463188, 1.54337969002848016527097969691, 2.70379324117270388106961852036, 3.94690575462706429459807059093, 4.95005816462521775049791746631, 5.90321134072117137282059336127, 6.59688514291544328488863509801, 7.47790906889535580210894405289, 8.313278373441961852812056575166, 9.112502595730814084580849886729

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