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Properties

Label 2-1200-240.227-c0-0-1
Degree $2$
Conductor $1200$
Sign $0.160 - 0.987i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 3-s − 4-s + i·6-s − i·8-s + 9-s − 12-s + 16-s + (−1 + i)17-s + i·18-s + (1 + i)19-s + (1 + i)23-s − i·24-s + 27-s − 2i·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s + 3-s − 4-s + i·6-s − i·8-s + 9-s − 12-s + 16-s + (−1 + i)17-s + i·18-s + (1 + i)19-s + (1 + i)23-s − i·24-s + 27-s − 2i·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 0.160 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.350641146\)
\(L(\frac12)\) \(\approx\) \(1.350641146\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.637450906837677117328221248887, −9.300539497930004516676687139493, −8.247735040832911389963267419658, −7.83577268762492082612155695589, −6.95283442066884362235293910954, −6.10109183576918176047941115420, −5.05842061467781748152997708081, −4.04498607870832072719601098739, −3.29278018354920551342566757020, −1.69184890118543220032359375778, 1.33429735571287586165829232487, 2.73250934799959549399788528511, 3.13171738102636819442293293872, 4.54017112006405775258489095369, 4.96134257722837535038927549667, 6.63549712264577725856802319743, 7.47034110889831128408493898925, 8.473306196265106933400171872393, 9.143732979727727472447007248396, 9.552912523416957621137522193713

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