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Properties

Label 2-1200-5.3-c2-0-7
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 + (6.02 − 6.02i)7-s + 2.99i·9-s − 17.0·11-s + (−0.124 − 0.124i)13-s + (−10.7 + 10.7i)17-s + 4.04i·19-s − 14.7·21-s + (1.85 + 1.85i)23-s + (3.67 − 3.67i)27-s + 54.4i·29-s + 53.1·31-s + (20.8 + 20.8i)33-s + (−21.2 + 21.2i)37-s + 0.305i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.860 − 0.860i)7-s + 0.333i·9-s − 1.54·11-s + (−0.00958 − 0.00958i)13-s + (−0.633 + 0.633i)17-s + 0.213i·19-s − 0.702·21-s + (0.0806 + 0.0806i)23-s + (0.136 − 0.136i)27-s + 1.87i·29-s + 1.71·31-s + (0.631 + 0.631i)33-s + (−0.574 + 0.574i)37-s + 0.00782i·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: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8756948559\)
\(L(\frac12)\) \(\approx\) \(0.8756948559\)
\(L(2)\) 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 \( 1 + (-6.02 + 6.02i)T - 49iT^{2} \)
11 \( 1 + 17.0T + 121T^{2} \)
13 \( 1 + (0.124 + 0.124i)T + 169iT^{2} \)
17 \( 1 + (10.7 - 10.7i)T - 289iT^{2} \)
19 \( 1 - 4.04iT - 361T^{2} \)
23 \( 1 + (-1.85 - 1.85i)T + 529iT^{2} \)
29 \( 1 - 54.4iT - 841T^{2} \)
31 \( 1 - 53.1T + 961T^{2} \)
37 \( 1 + (21.2 - 21.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 52.1T + 1.68e3T^{2} \)
43 \( 1 + (50.4 + 50.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-13.4 + 13.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (12.4 + 12.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 8.79iT - 3.48e3T^{2} \)
61 \( 1 - 105.T + 3.72e3T^{2} \)
67 \( 1 + (46.8 - 46.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 56.8T + 5.04e3T^{2} \)
73 \( 1 + (-52.7 - 52.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 15.2iT - 6.24e3T^{2} \)
83 \( 1 + (-46.1 - 46.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 0.948iT - 7.92e3T^{2} \)
97 \( 1 + (77.6 - 77.6i)T - 9.40e3iT^{2} \)
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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

−10.14450632711173207586699950801, −8.535181752673306843642246008042, −8.165099142608076381996339511063, −7.21229329229506907089188942612, −6.58198571450639023268364713562, −5.26207151307832121310621902586, −4.87244655773216320281798993147, −3.60599904485952279068392794343, −2.27990845187580873432009438712, −1.14277843696561215107834929528, 0.29200503110419599187992079312, 2.09701606936515163278855740889, 2.95541534172617621610148362977, 4.51340804441063748128739677411, 5.03928896380346460698850818611, 5.82772207876737993672029242977, 6.81916141812120987873819033894, 8.027494113692630692270804568435, 8.380289072790003564546207233461, 9.512549418728929512927171599581

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