Properties

Label 2-1200-12.11-c1-0-20
Degree $2$
Conductor $1200$
Sign $0.866 + 0.5i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (1.5 − 2.59i)9-s + 3·11-s − 2·13-s − 5.19i·17-s + 5.19i·19-s − 6·23-s + 5.19i·27-s − 10.3i·29-s − 3.46i·31-s + (−4.5 + 2.59i)33-s + 8·37-s + (3 − 1.73i)39-s − 5.19i·41-s − 3.46i·43-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.5 − 0.866i)9-s + 0.904·11-s − 0.554·13-s − 1.26i·17-s + 1.19i·19-s − 1.25·23-s + 0.999i·27-s − 1.92i·29-s − 0.622i·31-s + (−0.783 + 0.452i)33-s + 1.31·37-s + (0.480 − 0.277i)39-s − 0.811i·41-s − 0.528i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095209602\)
\(L(\frac12)\) \(\approx\) \(1.095209602\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + 5.19iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.754337560460811700069512717332, −9.181192802291979827371125520145, −7.940277479717745619180806268176, −7.13617777727614912949234822791, −6.11598497518027464190669059566, −5.60239882650989773031994577036, −4.37642736800182503572167199721, −3.86356699939317211641291694422, −2.29441613119590636210046084071, −0.63597144735035765118299971050, 1.10690793654110656542402701433, 2.30083713219875881569122642647, 3.82899820698980028168083646009, 4.76888859816786094262034067345, 5.69301132670595149418031678926, 6.55410115577573692881082713121, 7.11522854541445265490842928416, 8.120765976651224914736317122282, 8.966429718695992216810964432667, 9.976481671854491664169160539553

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