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Properties

Label 2-40e2-40.3-c1-0-9
Degree $2$
Conductor $1600$
Sign $-0.608 - 0.793i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087333517\)
\(L(\frac12)\) \(\approx\) \(1.087333517\)
\(L(\frac{3}{2})\) 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

−9.551274550653825394261810560133, −9.137284108714780069738153750897, −8.033736879761640800713805241051, −7.49376488242799113055783505343, −6.37406407197317584860592343172, −5.68025130083990437199169431854, −4.49806032612834087960524395576, −3.82573257081332299932683221298, −2.81181503810719868251474150437, −1.62274021450278194819523278949, 0.37369260456568433767436085558, 2.22963128507866059965801434032, 2.65734116129298140903283937713, 4.08224026612968912744766927003, 5.03022628325623371197984924074, 5.77827198684891359703138403274, 6.94985107098693936088386296135, 7.62086201465219278874705426008, 8.153557840070998017153600541681, 9.094823641478002037848812187826

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