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Properties

Label 2-1200-5.3-c2-0-29
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 + (4.77 − 4.77i)7-s + 2.99i·9-s − 20.6·11-s + (8.32 + 8.32i)13-s + (1.34 − 1.34i)17-s − 33.6i·19-s + 11.6·21-s + (−16.0 − 16.0i)23-s + (−3.67 + 3.67i)27-s − 8.69i·29-s + 49.0·31-s + (−25.3 − 25.3i)33-s + (36.4 − 36.4i)37-s + 20.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.682 − 0.682i)7-s + 0.333i·9-s − 1.88·11-s + (0.640 + 0.640i)13-s + (0.0793 − 0.0793i)17-s − 1.77i·19-s + 0.556·21-s + (−0.697 − 0.697i)23-s + (−0.136 + 0.136i)27-s − 0.299i·29-s + 1.58·31-s + (−0.768 − 0.768i)33-s + (0.986 − 0.986i)37-s + 0.522i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.749532684\)
\(L(\frac12)\) \(\approx\) \(1.749532684\)
\(L(2)\) 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 \( 1 + (-4.77 + 4.77i)T - 49iT^{2} \)
11 \( 1 + 20.6T + 121T^{2} \)
13 \( 1 + (-8.32 - 8.32i)T + 169iT^{2} \)
17 \( 1 + (-1.34 + 1.34i)T - 289iT^{2} \)
19 \( 1 + 33.6iT - 361T^{2} \)
23 \( 1 + (16.0 + 16.0i)T + 529iT^{2} \)
29 \( 1 + 8.69iT - 841T^{2} \)
31 \( 1 - 49.0T + 961T^{2} \)
37 \( 1 + (-36.4 + 36.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 33.3T + 1.68e3T^{2} \)
43 \( 1 + (24.1 + 24.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (13.9 - 13.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (59.3 + 59.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 30iT - 3.48e3T^{2} \)
61 \( 1 + 47.7T + 3.72e3T^{2} \)
67 \( 1 + (-17.0 + 17.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 9.30T + 5.04e3T^{2} \)
73 \( 1 + (22.2 + 22.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 27.3iT - 6.24e3T^{2} \)
83 \( 1 + (63.4 + 63.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 17.3iT - 7.92e3T^{2} \)
97 \( 1 + (41.8 - 41.8i)T - 9.40e3iT^{2} \)
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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

−9.444313156246940177542181028003, −8.424348132213703850185599705603, −7.927945427795179836801456260233, −7.08912835820192437878670240979, −5.98526892603589423436870097890, −4.77894314175557407919985425430, −4.43405315298708943616371734190, −3.03964902260561882728975641963, −2.16099170220949301263563291237, −0.49468915295161580528199684779, 1.32707819818830284044071281363, 2.45254583355364213559014455058, 3.30760494315131761246799542378, 4.65223401450562193236547567918, 5.62954624808088068899789018115, 6.17083204078803232568139894073, 7.76950268649223874506672780177, 7.936569792917960957090498465459, 8.578023147584300474472008201834, 9.855452856464278075278792444614

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