Properties

Label 2-1040-13.4-c1-0-16
Degree $2$
Conductor $1040$
Sign $0.979 - 0.203i$
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  + (−0.170 + 0.294i)3-s + i·5-s + (4.07 − 2.35i)7-s + (1.44 + 2.49i)9-s + (3.59 + 2.07i)11-s + (−2.36 − 2.72i)13-s + (−0.294 − 0.170i)15-s + (−2.86 − 4.96i)17-s + (5.05 − 2.91i)19-s + 1.60i·21-s + (−2.15 + 3.72i)23-s − 25-s − 2.00·27-s + (−4.09 + 7.08i)29-s − 4.89i·31-s + ⋯
L(s)  = 1  + (−0.0981 + 0.170i)3-s + 0.447i·5-s + (1.54 − 0.890i)7-s + (0.480 + 0.832i)9-s + (1.08 + 0.625i)11-s + (−0.656 − 0.754i)13-s + (−0.0760 − 0.0439i)15-s + (−0.695 − 1.20i)17-s + (1.16 − 0.669i)19-s + 0.349i·21-s + (−0.448 + 0.777i)23-s − 0.200·25-s − 0.385·27-s + (−0.759 + 1.31i)29-s − 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.979 - 0.203i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.979 - 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.985964278\)
\(L(\frac12)\) \(\approx\) \(1.985964278\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 + 2.72i)T \)
good3 \( 1 + (0.170 - 0.294i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-4.07 + 2.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.59 - 2.07i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.05 + 2.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.15 - 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 - 7.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + (-5.87 - 3.39i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.13 - 3.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.26 - 3.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + 3.11T + 53T^{2} \)
59 \( 1 + (1.36 - 0.790i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.06 - 3.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.27 - 4.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.708 + 0.408i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.75iT - 73T^{2} \)
79 \( 1 - 4.83T + 79T^{2} \)
83 \( 1 - 3.72iT - 83T^{2} \)
89 \( 1 + (-5.73 - 3.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.2 - 7.64i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.898953043604404330431488431926, −9.376524946911457284052216728137, −8.003592427367317513202931615009, −7.41198295392542090693537661029, −6.96081995496134437002455507257, −5.32905046997847627073597217372, −4.77049284483660739689131777970, −3.91726774137749923183877921614, −2.42158874035593457635661688574, −1.23892129856711769049882785497, 1.24504780730124118661046712105, 2.14750227758834157557109090363, 3.91025764413926993466223914408, 4.53661871546464261260934657211, 5.73476151876539336889520427821, 6.32570120282917727960889069734, 7.54208782286023083771837305863, 8.297450115005935174831782930075, 9.095706246833574706076718254082, 9.593641341692435826743336975785

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