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Properties

Label 2-40e2-80.43-c1-0-28
Degree $2$
Conductor $1600$
Sign $0.987 + 0.158i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + 2.35i·3-s + (2.66 − 2.66i)7-s − 2.56·9-s + (2.20 − 2.20i)11-s − 4.16·13-s + (1.69 − 1.69i)17-s + (4.74 − 4.74i)19-s + (6.28 + 6.28i)21-s + (−3.70 − 3.70i)23-s + 1.03i·27-s + (−3.65 − 3.65i)29-s − 6.90i·31-s + (5.20 + 5.20i)33-s + 1.10·37-s − 9.81i·39-s + ⋯
L(s)  = 1  + 1.36i·3-s + (1.00 − 1.00i)7-s − 0.853·9-s + (0.665 − 0.665i)11-s − 1.15·13-s + (0.410 − 0.410i)17-s + (1.08 − 1.08i)19-s + (1.37 + 1.37i)21-s + (−0.772 − 0.772i)23-s + 0.199i·27-s + (−0.679 − 0.679i)29-s − 1.23i·31-s + (0.905 + 0.905i)33-s + 0.180·37-s − 1.57i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.809012411\)
\(L(\frac12)\) \(\approx\) \(1.809012411\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 2.35iT - 3T^{2} \)
7 \( 1 + (-2.66 + 2.66i)T - 7iT^{2} \)
11 \( 1 + (-2.20 + 2.20i)T - 11iT^{2} \)
13 \( 1 + 4.16T + 13T^{2} \)
17 \( 1 + (-1.69 + 1.69i)T - 17iT^{2} \)
19 \( 1 + (-4.74 + 4.74i)T - 19iT^{2} \)
23 \( 1 + (3.70 + 3.70i)T + 23iT^{2} \)
29 \( 1 + (3.65 + 3.65i)T + 29iT^{2} \)
31 \( 1 + 6.90iT - 31T^{2} \)
37 \( 1 - 1.10T + 37T^{2} \)
41 \( 1 + 9.85iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (3.90 + 3.90i)T + 47iT^{2} \)
53 \( 1 - 6.19iT - 53T^{2} \)
59 \( 1 + (-3.42 - 3.42i)T + 59iT^{2} \)
61 \( 1 + (4.57 - 4.57i)T - 61iT^{2} \)
67 \( 1 + 6.37T + 67T^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 + (4.70 - 4.70i)T - 73iT^{2} \)
79 \( 1 - 2.54T + 79T^{2} \)
83 \( 1 - 7.65iT - 83T^{2} \)
89 \( 1 - 1.77T + 89T^{2} \)
97 \( 1 + (-1.16 + 1.16i)T - 97iT^{2} \)
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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

−9.463923253637165034480869552670, −8.861132532330522922514410630964, −7.67306666655502490178725546072, −7.27054858799421225041593261424, −5.87852763182062986599378191243, −5.01320529880585198593208698941, −4.34274164406258413385986387061, −3.72657404678946369697915540943, −2.46707487290048229756490427902, −0.74308132026571704710883863178, 1.48791848179874385895745144008, 1.90529360119557251669095058634, 3.17047376815015098766313928392, 4.60012853845603123989579601290, 5.48851165110627011979997546521, 6.19884781090443357302580446438, 7.28480969314318893629520078025, 7.68471219305150850159227677783, 8.379671908940773957759488824962, 9.386373925670282002609146168347

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