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Properties

Label 2-40e2-5.4-c1-0-15
Degree $2$
Conductor $1600$
Sign $0.447 - 0.894i$
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  + 2.82i·3-s − 2.82i·7-s − 5.00·9-s + 5.65·11-s − 2i·13-s + 2i·17-s + 8.00·21-s − 2.82i·23-s − 5.65i·27-s + 6·29-s + 5.65·31-s + 16.0i·33-s + 10i·37-s + 5.65·39-s + 2·41-s + ⋯
L(s)  = 1  + 1.63i·3-s − 1.06i·7-s − 1.66·9-s + 1.70·11-s − 0.554i·13-s + 0.485i·17-s + 1.74·21-s − 0.589i·23-s − 1.08i·27-s + 1.11·29-s + 1.01·31-s + 2.78i·33-s + 1.64i·37-s + 0.905·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.871897830\)
\(L(\frac12)\) \(\approx\) \(1.871897830\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82iT - 3T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2.82iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.865094262548910641723140412632, −8.789812767837845678232224479217, −8.324457671826309675838315156337, −6.99728077228937134649177493150, −6.30101724688792077497253102667, −5.20237943943613302465092369871, −4.22054382921406884478613210866, −3.96981655492187148013825241809, −2.92797579620847002424235102555, −1.03745201013272084248129158929, 1.00577800743495396839109662713, 1.96187683237837979839434404500, 2.86217587259847144809412136301, 4.18290299958026347422169719205, 5.45831158967915928654549026155, 6.33355625110433723899208867520, 6.71109540722337397623949502591, 7.58273783459155915124108526582, 8.477438332736358409645343428583, 9.046980860438293963395257908843

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