Properties

Label 2-1380-345.344-c1-0-22
Degree $2$
Conductor $1380$
Sign $0.982 - 0.187i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.779 − 1.54i)3-s + (0.268 + 2.21i)5-s + 2.49·7-s + (−1.78 − 2.41i)9-s + 3.17·11-s + 5.31i·13-s + (3.64 + 1.31i)15-s − 2.70i·17-s + 8.13i·19-s + (1.94 − 3.86i)21-s + (4.12 − 2.44i)23-s + (−4.85 + 1.19i)25-s + (−5.12 + 0.877i)27-s + 2.82i·29-s − 6.28·31-s + ⋯
L(s)  = 1  + (0.450 − 0.892i)3-s + (0.119 + 0.992i)5-s + 0.944·7-s + (−0.594 − 0.804i)9-s + 0.958·11-s + 1.47i·13-s + (0.940 + 0.339i)15-s − 0.655i·17-s + 1.86i·19-s + (0.425 − 0.843i)21-s + (0.859 − 0.510i)23-s + (−0.971 + 0.238i)25-s + (−0.985 + 0.168i)27-s + 0.524i·29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.982 - 0.187i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.982 - 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.277525467\)
\(L(\frac12)\) \(\approx\) \(2.277525467\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.779 + 1.54i)T \)
5 \( 1 + (-0.268 - 2.21i)T \)
23 \( 1 + (-4.12 + 2.44i)T \)
good7 \( 1 - 2.49T + 7T^{2} \)
11 \( 1 - 3.17T + 11T^{2} \)
13 \( 1 - 5.31iT - 13T^{2} \)
17 \( 1 + 2.70iT - 17T^{2} \)
19 \( 1 - 8.13iT - 19T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 6.28T + 31T^{2} \)
37 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 - 6.93T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + 3.79iT - 53T^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + 0.443iT - 61T^{2} \)
67 \( 1 + 6.68T + 67T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + 2.80iT - 73T^{2} \)
79 \( 1 + 6.08iT - 79T^{2} \)
83 \( 1 - 8.29iT - 83T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−9.299350792507749050072924886703, −8.898413822452752274979874067348, −7.75489090420807224883969674383, −7.23784864106524919226708325469, −6.49616636106170531663165096696, −5.73040894210443619419590792100, −4.30505608573284514535136580634, −3.43212551173657065292906082070, −2.18934738022653364157861843737, −1.45570511663442082348032853283, 1.00092736931986932886076514095, 2.40474500640577447688270042715, 3.63057643331641773868790038064, 4.55153319767701093437089344722, 5.14887104573350565943173144440, 5.94516988035533168383811135878, 7.43718232944490330983876644552, 8.117354449334830214418589043060, 8.973224776162388471966023915865, 9.243222784137207523385524418109

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