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Properties

Label 2-1040-13.4-c1-0-27
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: $\chi_{1040} (641, \cdot )$
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

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

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