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Properties

Label 2-1200-5.3-c2-0-10
Degree $2$
Conductor $1200$
Sign $-0.326 - 0.945i$
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 + (0.775 − 0.775i)7-s + 2.99i·9-s − 2.89·11-s + (−5.87 − 5.87i)13-s + (−4.44 + 4.44i)17-s + 0.101i·19-s + 1.89·21-s + (25.3 + 25.3i)23-s + (−3.67 + 3.67i)27-s + 32.2i·29-s + 3.69·31-s + (−3.55 − 3.55i)33-s + (−42.6 + 42.6i)37-s − 14.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.110 − 0.110i)7-s + 0.333i·9-s − 0.263·11-s + (−0.452 − 0.452i)13-s + (−0.261 + 0.261i)17-s + 0.00531i·19-s + 0.0904·21-s + (1.10 + 1.10i)23-s + (−0.136 + 0.136i)27-s + 1.11i·29-s + 0.119·31-s + (−0.107 − 0.107i)33-s + (−1.15 + 1.15i)37-s − 0.369i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.326 - 0.945i$
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: \((2,\ 1200,\ (\ :1),\ -0.326 - 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.611043262\)
\(L(\frac12)\) \(\approx\) \(1.611043262\)
\(L(2)\) 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 \( 1 + (-0.775 + 0.775i)T - 49iT^{2} \)
11 \( 1 + 2.89T + 121T^{2} \)
13 \( 1 + (5.87 + 5.87i)T + 169iT^{2} \)
17 \( 1 + (4.44 - 4.44i)T - 289iT^{2} \)
19 \( 1 - 0.101iT - 361T^{2} \)
23 \( 1 + (-25.3 - 25.3i)T + 529iT^{2} \)
29 \( 1 - 32.2iT - 841T^{2} \)
31 \( 1 - 3.69T + 961T^{2} \)
37 \( 1 + (42.6 - 42.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 12.8T + 1.68e3T^{2} \)
43 \( 1 + (-49.2 - 49.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-2.85 + 2.85i)T - 2.20e3iT^{2} \)
53 \( 1 + (-13.1 - 13.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 76.3iT - 3.48e3T^{2} \)
61 \( 1 + 103.T + 3.72e3T^{2} \)
67 \( 1 + (-47.6 + 47.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 29.7T + 5.04e3T^{2} \)
73 \( 1 + (3.50 + 3.50i)T + 5.32e3iT^{2} \)
79 \( 1 - 87.7iT - 6.24e3T^{2} \)
83 \( 1 + (-81.7 - 81.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 96.5iT - 7.92e3T^{2} \)
97 \( 1 + (54.2 - 54.2i)T - 9.40e3iT^{2} \)
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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

−9.687349875798514305558228338079, −9.092724484643605692537751981536, −8.170163021274610364162805778381, −7.47499003003155664664513310967, −6.55695155970147994131264109804, −5.35810381463356147161761194109, −4.73923680932013752025740713663, −3.54758579467082899553408122344, −2.74828943041257637993864849127, −1.36544559036328448732770326680, 0.45195040999602862800046453702, 1.98299829163897958726727449646, 2.82180184068461610375120443955, 4.05673746785782662863181541220, 4.99591650438561092047030415753, 6.02689953190962267791620593586, 7.01147265829155059642646927775, 7.54252464803321731250019188930, 8.664233257785018552253409253229, 9.054131470171303078950320635570

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