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Properties

Label 2-1040-13.10-c1-0-19
Degree $2$
Conductor $1040$
Sign $0.274 - 0.961i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (1.52 + 2.63i)3-s − i·5-s + (2.67 + 1.54i)7-s + (−3.12 + 5.41i)9-s + (5.36 − 3.09i)11-s + (2.54 + 2.55i)13-s + (2.63 − 1.52i)15-s + (3.24 − 5.61i)17-s + (−2.65 − 1.53i)19-s + 9.40i·21-s + (−0.896 − 1.55i)23-s − 25-s − 9.88·27-s + (−1.49 − 2.58i)29-s + 4.34i·31-s + ⋯
L(s)  = 1  + (0.878 + 1.52i)3-s − 0.447i·5-s + (1.01 + 0.584i)7-s + (−1.04 + 1.80i)9-s + (1.61 − 0.933i)11-s + (0.704 + 0.709i)13-s + (0.680 − 0.392i)15-s + (0.786 − 1.36i)17-s + (−0.609 − 0.351i)19-s + 2.05i·21-s + (−0.186 − 0.323i)23-s − 0.200·25-s − 1.90·27-s + (−0.276 − 0.479i)29-s + 0.780i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.274 - 0.961i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.274 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.614862850\)
\(L(\frac12)\) \(\approx\) \(2.614862850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (-2.54 - 2.55i)T \)
good3 \( 1 + (-1.52 - 2.63i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.67 - 1.54i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.36 + 3.09i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.24 + 5.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.65 + 1.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.896 + 1.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.34iT - 31T^{2} \)
37 \( 1 + (8.31 - 4.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.40 - 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.64 + 4.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 7.82T + 53T^{2} \)
59 \( 1 + (1.21 + 0.699i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.92 - 8.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.52 + 0.881i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.60 - 2.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.7iT - 73T^{2} \)
79 \( 1 + 6.58T + 79T^{2} \)
83 \( 1 - 4.46iT - 83T^{2} \)
89 \( 1 + (10.0 - 5.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.6 + 6.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.869228513942590883320794558076, −9.022819988286817079450576978634, −8.777997591018293361184650488462, −8.116240937963840872967661827515, −6.67367025839025956842660921692, −5.46460055046126492576163822392, −4.71599039135103205830725293340, −3.92680079007848261807970001261, −3.06386923485453471538636510226, −1.61765450099762429672020880744, 1.42426017487673592709458939895, 1.78935255490155753565446788957, 3.39176310793822523856882196825, 4.12226730614040065981566873892, 5.81865255208916245623909767135, 6.61882369865159537353390513690, 7.33546436616692458622775000199, 8.031896612903676769809543149312, 8.576171023712391098673765465016, 9.606624383820050666028883010314

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