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Properties

Label 2-1040-13.10-c1-0-15
Degree $2$
Conductor $1040$
Sign $0.999 + 0.0183i$
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  + (1.16 + 2.01i)3-s − i·5-s + (−3.11 − 1.80i)7-s + (−1.21 + 2.11i)9-s + (4.65 − 2.68i)11-s + (1.81 − 3.11i)13-s + (2.01 − 1.16i)15-s + (−0.565 + 0.980i)17-s + (1.96 + 1.13i)19-s − 8.39i·21-s + (1.94 + 3.37i)23-s − 25-s + 1.30·27-s + (0.0123 + 0.0214i)29-s − 5.46i·31-s + ⋯
L(s)  = 1  + (0.673 + 1.16i)3-s − 0.447i·5-s + (−1.17 − 0.680i)7-s + (−0.406 + 0.704i)9-s + (1.40 − 0.809i)11-s + (0.504 − 0.863i)13-s + (0.521 − 0.301i)15-s + (−0.137 + 0.237i)17-s + (0.450 + 0.260i)19-s − 1.83i·21-s + (0.405 + 0.702i)23-s − 0.200·25-s + 0.251·27-s + (0.00229 + 0.00397i)29-s − 0.981i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.999 + 0.0183i$
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: \((2,\ 1040,\ (\ :1/2),\ 0.999 + 0.0183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.942383042\)
\(L(\frac12)\) \(\approx\) \(1.942383042\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 3.11i)T \)
good3 \( 1 + (-1.16 - 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.11 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.565 - 0.980i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.94 - 3.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0123 - 0.0214i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (-7.53 + 4.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.565 + 0.980i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.58iT - 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (-0.148 - 0.0857i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.54 - 3.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.35 + 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.70iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.594164759312121362380847300675, −9.377145304376813318577602547161, −8.542628204629316727779906405243, −7.57371925715285597326462523773, −6.39169952698326378182242947474, −5.66269589651255508382253643343, −4.25285920828219330874700315385, −3.71682406450135677578932602425, −3.02802488954789968007773395411, −0.944037888400198958165819917902, 1.37883576292551375243655498180, 2.48752029831029824347764006305, 3.32948599485785575025110273723, 4.54593512997710727803370755626, 6.19778168103032290504131827645, 6.64390852329286995232982809062, 7.17925310285191947766709160196, 8.281613797461387566390687457980, 9.259025120276939386952370561801, 9.453992372171627293547615484406

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