Properties

Label 2-1200-15.14-c2-0-30
Degree $2$
Conductor $1200$
Sign $0.894 + 0.447i$
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  − 3i·3-s − 2i·7-s − 9·9-s + 22i·13-s + 26·19-s − 6·21-s + 27i·27-s + 46·31-s + 26i·37-s + 66·39-s − 22i·43-s + 45·49-s − 78i·57-s + 74·61-s + 18i·63-s + ⋯
L(s)  = 1  i·3-s − 0.285i·7-s − 9-s + 1.69i·13-s + 1.36·19-s − 0.285·21-s + i·27-s + 1.48·31-s + 0.702i·37-s + 1.69·39-s − 0.511i·43-s + 0.918·49-s − 1.36i·57-s + 1.21·61-s + 0.285i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 1200,\ (\ :1),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.849027619\)
\(L(\frac12)\) \(\approx\) \(1.849027619\)
\(L(2)\) 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 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 + 2iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 - 22iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 - 26T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 46T + 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 + 122iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 46iT - 5.32e3T^{2} \)
79 \( 1 + 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 2iT - 9.40e3T^{2} \)
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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

−9.379230643072970718762711422715, −8.624231399562789067105406458078, −7.73818475963099160195775975994, −6.97704308632355725154152325984, −6.40341970253319243018242083721, −5.35363652451805977746735975063, −4.30133988471074141101764734825, −3.10180569049129699416252087237, −1.97087687536586760557602191897, −0.912761532467478267400477492868, 0.74520079872564095509918015872, 2.67359654634481488137121858420, 3.35229523741576109504318034474, 4.47983358646155100819597314140, 5.43150070137522414738470990804, 5.89328273971928610717249220326, 7.26987969447683356154256588877, 8.162868764448231638176178227161, 8.811834134187798872657099847090, 9.880537737816700097522612741882

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