Properties

Label 2-40e2-1.1-c1-0-23
Degree $2$
Conductor $1600$
Sign $1$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s + 2.82·7-s + 5.00·9-s + 5.65·11-s − 2·13-s − 2·17-s + 8.00·21-s − 2.82·23-s + 5.65·27-s − 6·29-s + 5.65·31-s + 16.0·33-s − 10·37-s − 5.65·39-s + 2·41-s − 8.48·43-s + 2.82·47-s + 1.00·49-s − 5.65·51-s + 6·53-s − 11.3·59-s + 2·61-s + 14.1·63-s − 2.82·67-s − 8.00·69-s + 5.65·71-s + 6·73-s + ⋯
L(s)  = 1  + 1.63·3-s + 1.06·7-s + 1.66·9-s + 1.70·11-s − 0.554·13-s − 0.485·17-s + 1.74·21-s − 0.589·23-s + 1.08·27-s − 1.11·29-s + 1.01·31-s + 2.78·33-s − 1.64·37-s − 0.905·39-s + 0.312·41-s − 1.29·43-s + 0.412·47-s + 0.142·49-s − 0.792·51-s + 0.824·53-s − 1.47·59-s + 0.256·61-s + 1.78·63-s − 0.345·67-s − 0.963·69-s + 0.671·71-s + 0.702·73-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.581159247\)
\(L(\frac12)\) \(\approx\) \(3.581159247\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.217213684924187167006777694099, −8.639225088524735331753480486892, −7.990811881348407989967704229473, −7.23203599502431104428265703229, −6.41845644443075369845279756685, −5.02131327190238839039908654587, −4.13521943015852292143199079077, −3.46936882159321081901922128882, −2.20933294345756036163890187215, −1.52217646853372676400449939557, 1.52217646853372676400449939557, 2.20933294345756036163890187215, 3.46936882159321081901922128882, 4.13521943015852292143199079077, 5.02131327190238839039908654587, 6.41845644443075369845279756685, 7.23203599502431104428265703229, 7.990811881348407989967704229473, 8.639225088524735331753480486892, 9.217213684924187167006777694099

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