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Properties

Label 2-1040-13.10-c1-0-25
Degree $2$
Conductor $1040$
Sign $-0.659 + 0.751i$
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  + (1.19 + 2.06i)3-s − i·5-s + (−3.14 − 1.81i)7-s + (−1.33 + 2.31i)9-s + (−5.16 + 2.98i)11-s + (−1.20 − 3.39i)13-s + (2.06 − 1.19i)15-s + (−0.935 + 1.61i)17-s + (0.301 + 0.174i)19-s − 8.63i·21-s + (−3.66 − 6.34i)23-s − 25-s + 0.790·27-s + (−5.02 − 8.70i)29-s + 7.65i·31-s + ⋯
L(s)  = 1  + (0.687 + 1.19i)3-s − 0.447i·5-s + (−1.18 − 0.685i)7-s + (−0.444 + 0.770i)9-s + (−1.55 + 0.898i)11-s + (−0.333 − 0.942i)13-s + (0.532 − 0.307i)15-s + (−0.226 + 0.392i)17-s + (0.0692 + 0.0399i)19-s − 1.88i·21-s + (−0.763 − 1.32i)23-s − 0.200·25-s + 0.152·27-s + (−0.933 − 1.61i)29-s + 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.659 + 0.751i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1759793812\)
\(L(\frac12)\) \(\approx\) \(0.1759793812\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 3.39i)T \)
good3 \( 1 + (-1.19 - 2.06i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.14 + 1.81i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.16 - 2.98i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.935 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.301 - 0.174i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.66 + 6.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.02 + 8.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.65iT - 31T^{2} \)
37 \( 1 + (6.63 - 3.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.472 - 0.273i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.58 + 4.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 - 9.75T + 53T^{2} \)
59 \( 1 + (9.50 + 5.48i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.23 + 1.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.31 + 3.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.340iT - 73T^{2} \)
79 \( 1 + 6.43T + 79T^{2} \)
83 \( 1 + 7.69iT - 83T^{2} \)
89 \( 1 + (1.21 - 0.698i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.22 - 1.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.865776303301468300057608946887, −8.931682786043224140277828383006, −8.077191030193868535234799879763, −7.31837104324075428666385164661, −6.10355703414519921234112466694, −4.99917142807684641064491880644, −4.28857995064158355399084051655, −3.33530190499452420038539886566, −2.45455197369202445891134557804, −0.06494104212940235641583253764, 1.99650109803232058104758008020, 2.76684641452795877246475421826, 3.57562325310073065050718751670, 5.37653126784348590888493835585, 6.08989797913537000028113088711, 7.12323154965869324269799480345, 7.51654788304209499549492796254, 8.562427410550189691770952417972, 9.226096182163761126555972854025, 10.09902076057988228668595001425

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