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Properties

Label 2-1200-15.14-c2-0-17
Degree $2$
Conductor $1200$
Sign $0.841 - 0.540i$
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  + (−0.323 − 2.98i)3-s + 4.72i·7-s + (−8.79 + 1.92i)9-s − 4.76i·11-s − 1.06i·13-s − 26.7·17-s − 8.12·19-s + (14.0 − 1.52i)21-s + 40.0·23-s + (8.59 + 25.5i)27-s + 20.8i·29-s + 33.7·31-s + (−14.2 + 1.53i)33-s + 60.4i·37-s + (−3.18 + 0.344i)39-s + ⋯
L(s)  = 1  + (−0.107 − 0.994i)3-s + 0.675i·7-s + (−0.976 + 0.214i)9-s − 0.433i·11-s − 0.0820i·13-s − 1.57·17-s − 0.427·19-s + (0.671 − 0.0727i)21-s + 1.74·23-s + (0.318 + 0.948i)27-s + 0.719i·29-s + 1.08·31-s + (−0.430 + 0.0466i)33-s + 1.63i·37-s + (−0.0815 + 0.00884i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.841 - 0.540i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.273677739\)
\(L(\frac12)\) \(\approx\) \(1.273677739\)
\(L(2)\) 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 \( 1 + (0.323 + 2.98i)T \)
5 \( 1 \)
good7 \( 1 - 4.72iT - 49T^{2} \)
11 \( 1 + 4.76iT - 121T^{2} \)
13 \( 1 + 1.06iT - 169T^{2} \)
17 \( 1 + 26.7T + 289T^{2} \)
19 \( 1 + 8.12T + 361T^{2} \)
23 \( 1 - 40.0T + 529T^{2} \)
29 \( 1 - 20.8iT - 841T^{2} \)
31 \( 1 - 33.7T + 961T^{2} \)
37 \( 1 - 60.4iT - 1.36e3T^{2} \)
41 \( 1 + 59.2iT - 1.68e3T^{2} \)
43 \( 1 - 56.4iT - 1.84e3T^{2} \)
47 \( 1 + 9.68T + 2.20e3T^{2} \)
53 \( 1 + 93.1T + 2.80e3T^{2} \)
59 \( 1 + 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 57.7T + 3.72e3T^{2} \)
67 \( 1 - 101. iT - 4.48e3T^{2} \)
71 \( 1 - 90.1iT - 5.04e3T^{2} \)
73 \( 1 - 40.0iT - 5.32e3T^{2} \)
79 \( 1 - 65.3T + 6.24e3T^{2} \)
83 \( 1 - 117.T + 6.88e3T^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 - 15.2iT - 9.40e3T^{2} \)
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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

−9.373107110913436102232154743374, −8.649299267115493357092935254351, −8.155128713222667797740507573176, −6.88827653296091578882863187009, −6.54310948031050099158985066222, −5.50214224929811909006960025831, −4.64946801565734767575000486685, −3.10079398138074074206924932550, −2.31138695098583973213855356515, −1.03497852178170086435745717313, 0.44230964854628596309923360295, 2.25158739561207721520209621964, 3.42397597493224225411070434339, 4.44716971363723229415429477824, 4.86717163718109841053462526010, 6.16652188582614304759180638403, 6.89258728122670942665633455371, 7.931340381451737768718065164269, 8.939543427098064990443552420214, 9.407599457859322224385352575484

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