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Properties

Label 2-1200-12.11-c1-0-28
Degree $2$
Conductor $1200$
Sign $0.418 + 0.908i$
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 − 1.41i)3-s + (−1.00 − 2.82i)9-s + 4.89·11-s + 4.89·13-s − 3.46i·17-s + 3.46i·19-s − 6·23-s + (−5.00 − 1.41i)27-s + 2.82i·29-s + 3.46i·31-s + (4.89 − 6.92i)33-s + 4.89·37-s + (4.89 − 6.92i)39-s − 5.65i·41-s − 8.48i·43-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + (−0.333 − 0.942i)9-s + 1.47·11-s + 1.35·13-s − 0.840i·17-s + 0.794i·19-s − 1.25·23-s + (−0.962 − 0.272i)27-s + 0.525i·29-s + 0.622i·31-s + (0.852 − 1.20i)33-s + 0.805·37-s + (0.784 − 1.10i)39-s − 0.883i·41-s − 1.29i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.418 + 0.908i$
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: \((2,\ 1200,\ (\ :1/2),\ 0.418 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.213704210\)
\(L(\frac12)\) \(\approx\) \(2.213704210\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8.48iT - 67T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.300088566856951870299961509354, −8.797757008695829198108541694283, −8.024462490996297826290065075436, −7.08322716753657859101295343288, −6.39168010752464140424389693861, −5.64704405021879485678770775430, −4.02941466392208582516434478564, −3.47303397879361899317441194527, −2.05313929833207383003595654285, −1.04285687428598778156383408146, 1.46059831993586108228041325740, 2.83003784890277061352165408781, 4.06295508142864887722798714083, 4.23247609880593210699259738993, 5.83272815448742066084512827065, 6.35277375173126196220610173641, 7.66516393926671827890338660237, 8.445798256139168136763696865561, 9.113955539072418171591526180334, 9.726860798206249423817179432858

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