Properties

Label 2-1040-13.10-c1-0-22
Degree $2$
Conductor $1040$
Sign $-0.103 + 0.994i$
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  + (−0.275 − 0.477i)3-s i·5-s + (1.57 + 0.908i)7-s + (1.34 − 2.33i)9-s + (−0.759 + 0.438i)11-s + (−2.94 − 2.07i)13-s + (−0.477 + 0.275i)15-s + (2.47 − 4.28i)17-s + (−3.18 − 1.83i)19-s − 1.00i·21-s + (2.69 + 4.66i)23-s − 25-s − 3.14·27-s + (0.214 + 0.371i)29-s − 2.36i·31-s + ⋯
L(s)  = 1  + (−0.159 − 0.275i)3-s − 0.447i·5-s + (0.594 + 0.343i)7-s + (0.449 − 0.778i)9-s + (−0.229 + 0.132i)11-s + (−0.817 − 0.576i)13-s + (−0.123 + 0.0711i)15-s + (0.599 − 1.03i)17-s + (−0.729 − 0.421i)19-s − 0.218i·21-s + (0.562 + 0.973i)23-s − 0.200·25-s − 0.604·27-s + (0.0398 + 0.0690i)29-s − 0.424i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.103 + 0.994i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397381183\)
\(L(\frac12)\) \(\approx\) \(1.397381183\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.94 + 2.07i)T \)
good3 \( 1 + (0.275 + 0.477i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.57 - 0.908i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.759 - 0.438i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.47 + 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.18 + 1.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.69 - 4.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.214 - 0.371i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.36iT - 31T^{2} \)
37 \( 1 + (-7.40 + 4.27i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.03 + 0.600i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.00 - 1.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.92iT - 47T^{2} \)
53 \( 1 + 5.45T + 53T^{2} \)
59 \( 1 + (3.96 + 2.29i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.952 - 1.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.42 + 2.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.4 + 7.76i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 2.60iT - 83T^{2} \)
89 \( 1 + (-0.645 + 0.372i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.46 - 4.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.543165796983175664614550482133, −9.049282063468825955860159775546, −7.84698027299041008389871499378, −7.37905619069226572093472261210, −6.28244747792771087768971911743, −5.28412256457965263826324283193, −4.64310868832789374687087512437, −3.32953681441830716833729920623, −2.06886263005888142499309461867, −0.66089757776636581222029026397, 1.60546912566357794568833747138, 2.77251239090939786538350052790, 4.20561447742812891533708865635, 4.73912379024748217848318501022, 5.88024604099762592628398513914, 6.82182168197118648192814994347, 7.74893535722859888317105895278, 8.285478806939027066449381116099, 9.497075014641942648051271261608, 10.34544834993347655846895825315

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