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Properties

Label 2-1040-13.4-c1-0-19
Degree $2$
Conductor $1040$
Sign $0.823 + 0.566i$
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  + (0.0473 − 0.0820i)3-s − i·5-s + (4.18 − 2.41i)7-s + (1.49 + 2.59i)9-s + (0.926 + 0.534i)11-s + (0.331 − 3.59i)13-s + (−0.0820 − 0.0473i)15-s + (1.77 + 3.08i)17-s + (−4.96 + 2.86i)19-s − 0.457i·21-s + (3.54 − 6.13i)23-s − 25-s + 0.567·27-s + (−0.736 + 1.27i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (0.0273 − 0.0473i)3-s − 0.447i·5-s + (1.57 − 0.912i)7-s + (0.498 + 0.863i)9-s + (0.279 + 0.161i)11-s + (0.0918 − 0.995i)13-s + (−0.0211 − 0.0122i)15-s + (0.431 + 0.747i)17-s + (−1.13 + 0.657i)19-s − 0.0998i·21-s + (0.738 − 1.27i)23-s − 0.200·25-s + 0.109·27-s + (−0.136 + 0.236i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.823 + 0.566i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.037101776\)
\(L(\frac12)\) \(\approx\) \(2.037101776\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.331 + 3.59i)T \)
good3 \( 1 + (-0.0473 + 0.0820i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-4.18 + 2.41i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.926 - 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.77 - 3.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.54 + 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.736 - 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (0.0219 + 0.0126i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.232 + 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.77 + 3.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.51iT - 47T^{2} \)
53 \( 1 - 0.991T + 53T^{2} \)
59 \( 1 + (-7.55 + 4.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 2.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.72 + 3.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 8.78T + 79T^{2} \)
83 \( 1 - 0.725iT - 83T^{2} \)
89 \( 1 + (-11.6 - 6.75i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.97 - 1.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.18934819807773821451457698490, −8.733921176046226331708515747848, −8.091171932059941964924092094187, −7.64208301747696243852657825197, −6.53550090091826072452574337992, −5.26857993612112154243348658661, −4.65126183582442817148882653776, −3.82703299221522645853645333957, −2.09137019961252669713010675403, −1.12793892541397263598913092177, 1.42107357451831607992180990436, 2.50387617486155332064281552092, 3.85692412645512222937654749088, 4.76675449492443306803104571846, 5.68306574229458953534170998889, 6.72136328727074821328880795514, 7.42843218652499311579692611693, 8.523020710713891057128460996208, 9.083285331106060054895237905779, 9.874237102718531867277075718562

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