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Properties

Label 2-1040-13.10-c1-0-21
Degree $2$
Conductor $1040$
Sign $-0.184 + 0.982i$
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  + (−0.300 − 0.519i)3-s + i·5-s + (−1.24 − 0.719i)7-s + (1.31 − 2.28i)9-s + (2.40 − 1.38i)11-s + (−2.76 − 2.31i)13-s + (0.519 − 0.300i)15-s + (−2.25 + 3.90i)17-s + (3.75 + 2.16i)19-s + 0.863i·21-s + (−2.21 − 3.84i)23-s − 25-s − 3.38·27-s + (−3.93 − 6.81i)29-s − 4.16i·31-s + ⋯
L(s)  = 1  + (−0.173 − 0.300i)3-s + 0.447i·5-s + (−0.471 − 0.272i)7-s + (0.439 − 0.762i)9-s + (0.723 − 0.417i)11-s + (−0.767 − 0.641i)13-s + (0.134 − 0.0774i)15-s + (−0.546 + 0.945i)17-s + (0.860 + 0.496i)19-s + 0.188i·21-s + (−0.462 − 0.801i)23-s − 0.200·25-s − 0.651·27-s + (−0.730 − 1.26i)29-s − 0.747i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.184 + 0.982i$
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: \((2,\ 1040,\ (\ :1/2),\ -0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.137744447\)
\(L(\frac12)\) \(\approx\) \(1.137744447\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.76 + 2.31i)T \)
good3 \( 1 + (0.300 + 0.519i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.24 + 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.40 + 1.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.75 - 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.21 + 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.93 + 6.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.16iT - 31T^{2} \)
37 \( 1 + (-0.498 + 0.287i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.65 + 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.30 - 2.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.25 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.61 - 2.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.66iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + (5.15 - 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.37 - 0.793i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.915355742077025311569307174162, −8.926148533833487843341873070868, −7.902064540731770411549381103997, −7.08570514134852451347566520145, −6.33939924132447203030546091368, −5.66959857656895695934213156336, −4.13982502488043632114777435030, −3.50233599092802542806701080593, −2.12679333873173996622804463320, −0.53320814375665426426928773466, 1.54789635356810926434419728422, 2.82530006608832074194893197725, 4.18560837880352479974527154519, 4.88881154692080313502614669788, 5.71880248158609231246532873362, 7.05808376057403660466736353815, 7.37176808501853966856871696735, 8.756230415274221538681138899767, 9.461964742870084063018096068142, 9.869428316757847075544746628575

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