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Properties

Label 2-1040-65.29-c1-0-5
Degree $2$
Conductor $1040$
Sign $0.411 - 0.911i$
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  + (−1.86 + 1.07i)3-s + (−0.817 − 2.08i)5-s + (−2.54 − 1.46i)7-s + (0.817 − 1.41i)9-s + (−0.317 − 0.550i)11-s + (−3.60 − 0.0716i)13-s + (3.76 + 3.00i)15-s + (1.05 + 0.611i)17-s + (−0.682 + 1.18i)19-s + 6.32·21-s + (1.86 − 1.07i)23-s + (−3.66 + 3.40i)25-s − 2.93i·27-s + (1.5 + 2.59i)29-s + 8.96·31-s + ⋯
L(s)  = 1  + (−1.07 + 0.621i)3-s + (−0.365 − 0.930i)5-s + (−0.961 − 0.555i)7-s + (0.272 − 0.472i)9-s + (−0.0957 − 0.165i)11-s + (−0.999 − 0.0198i)13-s + (0.972 + 0.774i)15-s + (0.257 + 0.148i)17-s + (−0.156 + 0.271i)19-s + 1.38·21-s + (0.388 − 0.224i)23-s + (−0.732 + 0.680i)25-s − 0.565i·27-s + (0.278 + 0.482i)29-s + 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5267450639\)
\(L(\frac12)\) \(\approx\) \(0.5267450639\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.817 + 2.08i)T \)
13 \( 1 + (3.60 + 0.0716i)T \)
good3 \( 1 + (1.86 - 1.07i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.54 + 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.317 + 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.05 - 0.611i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.86 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + (-1.05 + 0.611i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.18 - 0.683i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.95 - 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (6.27 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.8 - 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.00171460007198642498516700409, −9.587129800334239425090855087734, −8.451644185418648866811981807713, −7.56180308700050022780836995156, −6.51306101359274552743839891851, −5.71480451907738887382553214505, −4.76461509581359624672787751621, −4.23794461263930432204235071092, −2.93047262022320151326568229657, −0.861264723354494521898680042382, 0.38092368172820071579314715862, 2.36643465732124748367853660498, 3.27466293185922342458617306613, 4.67580018125466694108473376107, 5.75958057298460471009280711976, 6.42479371484608402723059629219, 7.05276704800474492450021204457, 7.78747215447272362578898837850, 9.080173476620707750562935855514, 9.953501285824748127772971190568

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