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Properties

Label 2-1040-13.4-c1-0-12
Degree $2$
Conductor $1040$
Sign $0.265 - 0.964i$
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  + (−1.36 + 2.36i)3-s + i·5-s + (2.59 − 1.5i)7-s + (−2.23 − 3.86i)9-s + (2.59 + 1.5i)11-s + (3.5 − 0.866i)13-s + (−2.36 − 1.36i)15-s + (1.09 + 1.90i)17-s + (5.59 − 3.23i)19-s + 8.19i·21-s + (1.26 − 2.19i)23-s − 25-s + 4.00·27-s + (4.73 − 8.19i)29-s + 1.26i·31-s + ⋯
L(s)  = 1  + (−0.788 + 1.36i)3-s + 0.447i·5-s + (0.981 − 0.566i)7-s + (−0.744 − 1.28i)9-s + (0.783 + 0.452i)11-s + (0.970 − 0.240i)13-s + (−0.610 − 0.352i)15-s + (0.266 + 0.461i)17-s + (1.28 − 0.741i)19-s + 1.78i·21-s + (0.264 − 0.457i)23-s − 0.200·25-s + 0.769·27-s + (0.878 − 1.52i)29-s + 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.265 - 0.964i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.551046867\)
\(L(\frac12)\) \(\approx\) \(1.551046867\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 0.866i)T \)
good3 \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.09 - 1.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.59 + 3.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.26 + 2.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.73 + 8.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (-9.69 - 5.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9 + 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (9 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.09 - 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.19 - 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 + 6.19T + 79T^{2} \)
83 \( 1 - 2.19iT - 83T^{2} \)
89 \( 1 + (14.8 + 8.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.0 + 7.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15987811023283978849507407445, −9.580005749245171286503906953692, −8.556763711438361140754711848438, −7.61713408604955068751061709918, −6.52253756228182231270517041633, −5.72229010173481100052549280894, −4.65470650329626435820601175594, −4.20556699971058825703679206421, −3.10560466506255441381668625812, −1.17443018344629294372952734912, 1.09723066898330222835607086177, 1.66400554434965065468869114701, 3.32783945284583886057813264868, 4.83086463081268090677316218374, 5.62337398653843095869496329770, 6.27950395794813015520817028944, 7.23389117786401566016633038222, 8.028627270007906174221050561792, 8.697098175154024836062850145092, 9.626969881898671269037701194163

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