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Properties

Label 2-40e2-40.27-c1-0-7
Degree $2$
Conductor $1600$
Sign $-0.382 - 0.923i$
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  + (−0.707 + 0.707i)3-s + 1.99i·9-s + 1.73·11-s + (2.82 + 2.82i)13-s + (−1.22 − 1.22i)17-s + 5.19i·19-s + (−4.89 − 4.89i)23-s + (−3.53 − 3.53i)27-s + 6.92·29-s − 4i·31-s + (−1.22 + 1.22i)33-s + (−5.65 + 5.65i)37-s − 4.00·39-s + 3·41-s + (−5.65 + 5.65i)43-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + 0.666i·9-s + 0.522·11-s + (0.784 + 0.784i)13-s + (−0.297 − 0.297i)17-s + 1.19i·19-s + (−1.02 − 1.02i)23-s + (−0.680 − 0.680i)27-s + 1.28·29-s − 0.718i·31-s + (−0.213 + 0.213i)33-s + (−0.929 + 0.929i)37-s − 0.640·39-s + 0.468·41-s + (−0.862 + 0.862i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219887309\)
\(L(\frac12)\) \(\approx\) \(1.219887309\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + 17iT^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (4.89 + 4.89i)T + 23iT^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (5.65 - 5.65i)T - 37iT^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (-4.89 + 4.89i)T - 47iT^{2} \)
53 \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (4.94 + 4.94i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (6.12 - 6.12i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 3iT - 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.888138011829028749727984458184, −8.735155210672627422920302622228, −8.261425108835788191976865413006, −7.19415241444524821205014084566, −6.27410411663912612375961204904, −5.68474214831264826871796533582, −4.45197464767543214754913032764, −4.07592872819554934550152021802, −2.63664985203648139816597111801, −1.42838577536204313411444097810, 0.52587035887760909966628918424, 1.73524447107787891214306983391, 3.19527766147999966779548822191, 3.98041145897093721291724668118, 5.18833991148908023689652161330, 6.00565269769813463350540684999, 6.69244780946611109603682547232, 7.39981695496466471264079431099, 8.515938668734861799102336933105, 8.996513645720774300082880430682

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