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Properties

Label 2-1040-13.10-c1-0-26
Degree $2$
Conductor $1040$
Sign $-0.991 - 0.129i$
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  + (−0.653 − 1.13i)3-s − i·5-s + (−2.51 − 1.45i)7-s + (0.646 − 1.11i)9-s + (−0.174 + 0.100i)11-s + (2.15 − 2.89i)13-s + (−1.13 + 0.653i)15-s + (−1.53 + 2.65i)17-s + (0.0122 + 0.00708i)19-s + 3.79i·21-s + (−2.36 − 4.09i)23-s − 25-s − 5.60·27-s + (1.05 + 1.82i)29-s + 1.79i·31-s + ⋯
L(s)  = 1  + (−0.377 − 0.653i)3-s − 0.447i·5-s + (−0.950 − 0.548i)7-s + (0.215 − 0.373i)9-s + (−0.0526 + 0.0303i)11-s + (0.597 − 0.801i)13-s + (−0.292 + 0.168i)15-s + (−0.371 + 0.643i)17-s + (0.00281 + 0.00162i)19-s + 0.828i·21-s + (−0.493 − 0.854i)23-s − 0.200·25-s − 1.07·27-s + (0.196 + 0.339i)29-s + 0.322i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.991 - 0.129i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6524704478\)
\(L(\frac12)\) \(\approx\) \(0.6524704478\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 + 2.89i)T \)
good3 \( 1 + (0.653 + 1.13i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.51 + 1.45i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.174 - 0.100i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.53 - 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0122 - 0.00708i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.05 - 1.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.79iT - 31T^{2} \)
37 \( 1 + (0.503 - 0.290i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.87 - 3.97i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.63 - 8.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.37iT - 47T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 + (5.03 + 2.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.95 - 3.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.80 + 1.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.65 - 3.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 15.2iT - 83T^{2} \)
89 \( 1 + (2.67 - 1.54i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.9 + 9.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.608898858293128951227897435401, −8.551077393768421991165540644065, −7.85022402777348952455528542395, −6.61440855204727151239083234513, −6.44384244496555393547171226230, −5.28374536716896280571421729952, −4.07286630794049325380781776019, −3.18377397278528037972147927348, −1.53987907489971994970680853719, −0.30694423187232295115964395544, 2.02292921023260509557765398131, 3.25805393983544838161511705121, 4.17363325957289566599996432197, 5.22045408396655154497447649678, 6.10416080912068451884121618479, 6.84602675084731772480672619215, 7.82685301304226523592093807632, 8.985189784834266440627431118387, 9.598603567560180058119660744018, 10.27507480139475169689748920364

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