Properties

Label 2-1200-15.14-c2-0-39
Degree $2$
Conductor $1200$
Sign $0.719 - 0.694i$
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  + (2.82 + i)3-s + 6i·7-s + (7.00 + 5.65i)9-s − 5.65i·11-s − 10i·13-s + 22.6·17-s + 2·19-s + (−6 + 16.9i)21-s + 11.3·23-s + (14.1 + 23.0i)27-s − 16.9i·29-s + 22·31-s + (5.65 − 16.0i)33-s − 6i·37-s + (10 − 28.2i)39-s + ⋯
L(s)  = 1  + (0.942 + 0.333i)3-s + 0.857i·7-s + (0.777 + 0.628i)9-s − 0.514i·11-s − 0.769i·13-s + 1.33·17-s + 0.105·19-s + (−0.285 + 0.808i)21-s + 0.491·23-s + (0.523 + 0.851i)27-s − 0.585i·29-s + 0.709·31-s + (0.171 − 0.484i)33-s − 0.162i·37-s + (0.256 − 0.725i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.719 - 0.694i$
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: \((2,\ 1200,\ (\ :1),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.016654461\)
\(L(\frac12)\) \(\approx\) \(3.016654461\)
\(L(2)\) 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 \( 1 + (-2.82 - i)T \)
5 \( 1 \)
good7 \( 1 - 6iT - 49T^{2} \)
11 \( 1 + 5.65iT - 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 - 22.6T + 289T^{2} \)
19 \( 1 - 2T + 361T^{2} \)
23 \( 1 - 11.3T + 529T^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 - 22T + 961T^{2} \)
37 \( 1 + 6iT - 1.36e3T^{2} \)
41 \( 1 - 33.9iT - 1.68e3T^{2} \)
43 \( 1 - 82iT - 1.84e3T^{2} \)
47 \( 1 - 67.8T + 2.20e3T^{2} \)
53 \( 1 + 62.2T + 2.80e3T^{2} \)
59 \( 1 - 73.5iT - 3.48e3T^{2} \)
61 \( 1 + 86T + 3.72e3T^{2} \)
67 \( 1 + 2iT - 4.48e3T^{2} \)
71 \( 1 - 124. iT - 5.04e3T^{2} \)
73 \( 1 + 82iT - 5.32e3T^{2} \)
79 \( 1 - 10T + 6.24e3T^{2} \)
83 \( 1 - 73.5T + 6.88e3T^{2} \)
89 \( 1 - 33.9iT - 7.92e3T^{2} \)
97 \( 1 + 94iT - 9.40e3T^{2} \)
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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

−9.575181835008413321440826966039, −8.848638624078785071083263181555, −8.054436853776840696576780057665, −7.55149664404458741229586110050, −6.18654984821441624099324595274, −5.41632925145245739796873318481, −4.40779874694808287284868531541, −3.18881363189275794261098118281, −2.69425762212409469461542112496, −1.20575324957054945883953449563, 0.947275998151289551516037014993, 2.02869926988907983939862029892, 3.27662824567679045423743814768, 4.03068789230044918146839830424, 5.03814207501648209222907801610, 6.37114532820351870517863009458, 7.24978780252556409182388607985, 7.63402198411879249270306355354, 8.678861478107384049022290999875, 9.391315447326588680859090236986

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