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Properties

Label 2-1040-13.4-c1-0-6
Degree $2$
Conductor $1040$
Sign $-0.343 - 0.939i$
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  + (−1.00 + 1.73i)3-s − i·5-s + (−1.48 + 0.858i)7-s + (−0.511 − 0.885i)9-s + (4.59 + 2.65i)11-s + (2.34 − 2.73i)13-s + (1.73 + 1.00i)15-s + (0.811 + 1.40i)17-s + (−1.96 + 1.13i)19-s − 3.44i·21-s + (2.52 − 4.37i)23-s − 25-s − 3.96·27-s + (−2.08 + 3.61i)29-s + 8.79i·31-s + ⋯
L(s)  = 1  + (−0.578 + 1.00i)3-s − 0.447i·5-s + (−0.561 + 0.324i)7-s + (−0.170 − 0.295i)9-s + (1.38 + 0.799i)11-s + (0.651 − 0.758i)13-s + (0.448 + 0.258i)15-s + (0.196 + 0.340i)17-s + (−0.451 + 0.260i)19-s − 0.751i·21-s + (0.526 − 0.912i)23-s − 0.200·25-s − 0.763·27-s + (−0.387 + 0.671i)29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.189605930\)
\(L(\frac12)\) \(\approx\) \(1.189605930\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 + 2.73i)T \)
good3 \( 1 + (1.00 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.48 - 0.858i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.59 - 2.65i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.811 - 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 + 4.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.08 - 3.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.79iT - 31T^{2} \)
37 \( 1 + (0.942 + 0.544i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.86 - 4.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.18 - 7.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.45iT - 47T^{2} \)
53 \( 1 + 5.54T + 53T^{2} \)
59 \( 1 + (8.57 - 4.94i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.373 - 0.646i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.5 + 7.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.13 - 1.23i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 0.702T + 79T^{2} \)
83 \( 1 + 3.57iT - 83T^{2} \)
89 \( 1 + (-9.93 - 5.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 7.03i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.26735527480437320808062115944, −9.292668829587693926651418329376, −8.914834073655479208806857224460, −7.73365879768313362271192587322, −6.53153296478828007646307677290, −5.89645295287262134107812054454, −4.83708607927485234518442793022, −4.18409421939388440667463837811, −3.16539694406742194767343617160, −1.40193820636089949636065430394, 0.64241284029492849360168968923, 1.85311969931443936611856575249, 3.40319331770347010423500268241, 4.19061056328892234907939417646, 5.92341317136450272862188050840, 6.22950367118638075701472598473, 7.04102861358059402273358836972, 7.69224488711904234118800533860, 9.041326328206940483101636911170, 9.463022305688999064151460009625

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