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Properties

Label 2-1040-65.29-c1-0-16
Degree $2$
Conductor $1040$
Sign $0.765 - 0.643i$
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  + (0.298 − 0.172i)3-s + (1.44 − 1.71i)5-s + (1.75 + 1.01i)7-s + (−1.44 + 2.49i)9-s + (1.94 + 3.36i)11-s + (−2.96 + 2.05i)13-s + (0.135 − 0.759i)15-s + (4.71 + 2.72i)17-s + (−2.94 + 5.09i)19-s + 0.700·21-s + (−0.298 + 0.172i)23-s + (−0.850 − 4.92i)25-s + 2.02i·27-s + (1.5 + 2.59i)29-s − 1.18·31-s + ⋯
L(s)  = 1  + (0.172 − 0.0996i)3-s + (0.644 − 0.764i)5-s + (0.664 + 0.383i)7-s + (−0.480 + 0.831i)9-s + (0.585 + 1.01i)11-s + (−0.821 + 0.570i)13-s + (0.0349 − 0.196i)15-s + (1.14 + 0.660i)17-s + (−0.674 + 1.16i)19-s + 0.152·21-s + (−0.0623 + 0.0359i)23-s + (−0.170 − 0.985i)25-s + 0.390i·27-s + (0.278 + 0.482i)29-s − 0.212·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.765 - 0.643i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.929662362\)
\(L(\frac12)\) \(\approx\) \(1.929662362\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 1.71i)T \)
13 \( 1 + (2.96 - 2.05i)T \)
good3 \( 1 + (-0.298 + 0.172i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.75 - 1.01i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.71 - 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 - 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.298 - 0.172i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + (-4.71 + 2.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0902 + 0.156i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.15 - 0.669i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 + 2.42iT - 53T^{2} \)
59 \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.81 + 2.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.940 - 1.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 7.83iT - 83T^{2} \)
89 \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.937287991220982648466762264606, −9.197149504685670422704954569903, −8.304044710720560787434007985614, −7.77452998606469192325128763762, −6.60407591094551643100424145589, −5.52399549807158486904106319058, −4.96125381602111622020215200859, −3.94362464021249001622412147063, −2.21744156692565254759346125640, −1.65670644237685268914681205620, 0.900655008674030558555509546236, 2.61548845689197633152467966488, 3.30347880054143750647475632118, 4.57473419638852680242808449532, 5.69053339288254986579733470390, 6.37472159874274873587484621549, 7.32710067421084878473557545109, 8.153093715916831889115686445256, 9.180499814658667682413000047684, 9.731870371347539929573550464868

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