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Properties

Label 2-1040-13.10-c1-0-16
Degree $2$
Conductor $1040$
Sign $0.848 + 0.529i$
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  + (0.249 + 0.431i)3-s + i·5-s + (−1.15 − 0.669i)7-s + (1.37 − 2.38i)9-s + (1.97 − 1.14i)11-s + (3.16 − 1.73i)13-s + (−0.431 + 0.249i)15-s + (0.209 − 0.363i)17-s + (−5.31 − 3.06i)19-s − 0.666i·21-s + (−3.05 − 5.28i)23-s − 25-s + 2.86·27-s + (2.40 + 4.16i)29-s + 9.98i·31-s + ⋯
L(s)  = 1  + (0.143 + 0.249i)3-s + 0.447i·5-s + (−0.438 − 0.252i)7-s + (0.458 − 0.794i)9-s + (0.595 − 0.344i)11-s + (0.876 − 0.481i)13-s + (−0.111 + 0.0643i)15-s + (0.0508 − 0.0881i)17-s + (−1.21 − 0.704i)19-s − 0.145i·21-s + (−0.636 − 1.10i)23-s − 0.200·25-s + 0.551·27-s + (0.446 + 0.774i)29-s + 1.79i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.848 + 0.529i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.848 + 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.670811290\)
\(L(\frac12)\) \(\approx\) \(1.670811290\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.16 + 1.73i)T \)
good3 \( 1 + (-0.249 - 0.431i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.15 + 0.669i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.97 + 1.14i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.209 + 0.363i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.31 + 3.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.05 + 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.40 - 4.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.98iT - 31T^{2} \)
37 \( 1 + (-9.45 + 5.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.41 + 5.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.50iT - 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 + (-6.99 - 4.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.338 - 0.195i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.4 - 6.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 + 7.35iT - 83T^{2} \)
89 \( 1 + (7.80 - 4.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.3 + 5.97i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.931169692612124188522807324299, −8.915996845370852767672060842227, −8.485549369614366708990161056079, −7.07137769840337453515772423861, −6.58714226258762432288547863347, −5.74361170758738758777027261480, −4.27017392531220174214410664061, −3.68331579196846331173614414798, −2.58265291085773581979056528067, −0.840298752998936863693738888438, 1.38151700929034570314919546538, 2.42978657825078629434180837875, 3.97688746562472658120736623010, 4.53527742116609357922569301420, 6.04580026162263038128210548406, 6.35289419192326945043829630583, 7.88585730213309922039222738588, 8.023548599462973494291658224469, 9.445043424815894935278380928354, 9.642290827507077975169951141700

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