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Properties

Label 2-1040-13.10-c1-0-4
Degree $2$
Conductor $1040$
Sign $-0.994 + 0.103i$
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  + (1.62 + 2.81i)3-s + i·5-s + (−4.00 − 2.31i)7-s + (−3.77 + 6.53i)9-s + (2.79 − 1.61i)11-s + (−1.42 + 3.31i)13-s + (−2.81 + 1.62i)15-s + (−2.77 + 4.81i)17-s + (−1.72 − 0.996i)19-s − 15.0i·21-s + (−2.20 − 3.81i)23-s − 25-s − 14.7·27-s + (−0.197 − 0.342i)29-s + 7.20i·31-s + ⋯
L(s)  = 1  + (0.937 + 1.62i)3-s + 0.447i·5-s + (−1.51 − 0.874i)7-s + (−1.25 + 2.17i)9-s + (0.843 − 0.486i)11-s + (−0.395 + 0.918i)13-s + (−0.726 + 0.419i)15-s + (−0.673 + 1.16i)17-s + (−0.395 − 0.228i)19-s − 3.27i·21-s + (−0.459 − 0.795i)23-s − 0.200·25-s − 2.84·27-s + (−0.0367 − 0.0636i)29-s + 1.29i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.192391510\)
\(L(\frac12)\) \(\approx\) \(1.192391510\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 3.31i)T \)
good3 \( 1 + (-1.62 - 2.81i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (4.00 + 2.31i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.79 + 1.61i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.77 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.72 + 0.996i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.20 + 3.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.197 + 0.342i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.20iT - 31T^{2} \)
37 \( 1 + (-0.708 + 0.408i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.79 - 1.61i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.42 - 5.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.19iT - 47T^{2} \)
53 \( 1 + 0.512T + 53T^{2} \)
59 \( 1 + (-6.82 - 3.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.08 + 5.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.58 + 4.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.43 - 1.40i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.03iT - 73T^{2} \)
79 \( 1 - 8.25T + 79T^{2} \)
83 \( 1 - 1.49iT - 83T^{2} \)
89 \( 1 + (-14.1 + 8.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.84 - 2.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31247857089866128537500065571, −9.542008875914444450508791758404, −8.958340997481093607050440562671, −8.191776434001532971270344265743, −6.80697269673961622690041722566, −6.31758260700697997362420735648, −4.77074332364965172938951799829, −3.83516861555845263484554042780, −3.55341446308770598506161714628, −2.36201059887819938313093607028, 0.44304894941775728190220997553, 2.01995275234241434370369402747, 2.79914412483910776465036341493, 3.76289616041001396728310311365, 5.50209474909932610035598183548, 6.36182879165390556652010797057, 6.97480803184445072136242708708, 7.79590226999255095331135104063, 8.682944888903903553166516749112, 9.352370944208864614248705212256

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