Properties

Label 2-1200-5.2-c2-0-19
Degree $2$
Conductor $1200$
Sign $0.229 + 0.973i$
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 + (−5.44 − 5.44i)7-s − 2.99i·9-s + 6.44·11-s + (−14.4 + 14.4i)13-s + (23.1 + 23.1i)17-s − 16.6i·19-s + 13.3·21-s + (−6.65 + 6.65i)23-s + (3.67 + 3.67i)27-s − 0.0454i·29-s − 4.49·31-s + (−7.89 + 7.89i)33-s + (−35.3 − 35.3i)37-s − 35.3i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.778 − 0.778i)7-s − 0.333i·9-s + 0.586·11-s + (−1.11 + 1.11i)13-s + (1.36 + 1.36i)17-s − 0.878i·19-s + 0.635·21-s + (−0.289 + 0.289i)23-s + (0.136 + 0.136i)27-s − 0.00156i·29-s − 0.144·31-s + (−0.239 + 0.239i)33-s + (−0.955 − 0.955i)37-s − 0.907i·39-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9134837520\)
\(L(\frac12)\) \(\approx\) \(0.9134837520\)
\(L(2)\) 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 \( 1 + (5.44 + 5.44i)T + 49iT^{2} \)
11 \( 1 - 6.44T + 121T^{2} \)
13 \( 1 + (14.4 - 14.4i)T - 169iT^{2} \)
17 \( 1 + (-23.1 - 23.1i)T + 289iT^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
23 \( 1 + (6.65 - 6.65i)T - 529iT^{2} \)
29 \( 1 + 0.0454iT - 841T^{2} \)
31 \( 1 + 4.49T + 961T^{2} \)
37 \( 1 + (35.3 + 35.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 20.2T + 1.68e3T^{2} \)
43 \( 1 + (-32.2 + 32.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (50.5 + 50.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (-5.50 + 5.50i)T - 2.80e3iT^{2} \)
59 \( 1 + 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (85.2 + 85.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (-21.9 + 21.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-94.9 + 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (-37 - 37i)T + 9.40e3iT^{2} \)
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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

−9.568082905246422101631492218790, −8.772746694425125187325223976670, −7.52462636110374345270550294838, −6.88116028216464584130233800025, −6.08887138132199645683454806145, −5.06699503737823611060079944460, −4.05552546070067765369016699479, −3.43134904005832387549738829898, −1.84710467920906373025609145796, −0.33663142202162034631448183360, 1.02687571808883546921744509434, 2.57274028360478869410010828879, 3.32135062171533998260440095972, 4.81615609996670574802233513871, 5.62965163750898561125636311123, 6.26705047671334290082017060210, 7.33819782001210578395206669562, 7.890847307584583181184697379434, 9.024916255594406796108727393402, 9.829752424318583896917928037292

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