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Properties

Label 2-1200-5.2-c2-0-0
Degree $2$
Conductor $1200$
Sign $-0.973 + 0.229i$
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 + (2.44 + 2.44i)7-s − 2.99i·9-s − 6·11-s + (12.2 − 12.2i)13-s + (−14.6 − 14.6i)17-s + 10i·19-s − 5.99·21-s + (−29.3 + 29.3i)23-s + (3.67 + 3.67i)27-s + 48i·29-s + 26·31-s + (7.34 − 7.34i)33-s + (−31.8 − 31.8i)37-s + 29.9i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.349 + 0.349i)7-s − 0.333i·9-s − 0.545·11-s + (0.942 − 0.942i)13-s + (−0.864 − 0.864i)17-s + 0.526i·19-s − 0.285·21-s + (−1.27 + 1.27i)23-s + (0.136 + 0.136i)27-s + 1.65i·29-s + 0.838·31-s + (0.222 − 0.222i)33-s + (−0.860 − 0.860i)37-s + 0.769i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
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: \((2,\ 1200,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1296429172\)
\(L(\frac12)\) \(\approx\) \(0.1296429172\)
\(L(2)\) 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 \( 1 + (-2.44 - 2.44i)T + 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \)
17 \( 1 + (14.6 + 14.6i)T + 289iT^{2} \)
19 \( 1 - 10iT - 361T^{2} \)
23 \( 1 + (29.3 - 29.3i)T - 529iT^{2} \)
29 \( 1 - 48iT - 841T^{2} \)
31 \( 1 - 26T + 961T^{2} \)
37 \( 1 + (31.8 + 31.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 30T + 1.68e3T^{2} \)
43 \( 1 + (29.3 - 29.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (14.6 + 14.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (14.6 - 14.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 78iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (63.6 + 63.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 120T + 5.04e3T^{2} \)
73 \( 1 + (83.2 - 83.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 74iT - 6.24e3T^{2} \)
83 \( 1 + (-44.0 + 44.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 150iT - 7.92e3T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 9.40e3iT^{2} \)
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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

−10.10271974560166852906537638705, −9.195179595036780040708104132602, −8.385061462769838098557455856803, −7.64017561445916535941620653640, −6.55407344131615032576316586103, −5.61145443792083203660057313036, −5.08248395027398204444576752464, −3.91443760724332416620707391183, −2.95335267150158216090052527296, −1.54817453605242963864031585183, 0.03997474052942285267603031522, 1.50096540507825417849013386941, 2.53269573771362903882498962722, 4.11668315582681139257306786202, 4.63048264281876853914591475028, 6.07171337603881759524851831416, 6.39542026536288844496662099382, 7.48383471265257630517567814060, 8.308478889419562867492987727200, 8.911588173490415322673812991793

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