Properties

Label 2-40e2-20.3-c1-0-22
Degree $2$
Conductor $1600$
Sign $0.525 + 0.850i$
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  + 3i·9-s + (−5 − 5i)13-s + (5 − 5i)17-s − 4i·29-s + (5 − 5i)37-s + 8·41-s − 7i·49-s + (5 + 5i)53-s + 12·61-s + (−5 − 5i)73-s − 9·81-s − 16i·89-s + (−5 + 5i)97-s − 2·101-s − 6i·109-s + ⋯
L(s)  = 1  + i·9-s + (−1.38 − 1.38i)13-s + (1.21 − 1.21i)17-s − 0.742i·29-s + (0.821 − 0.821i)37-s + 1.24·41-s i·49-s + (0.686 + 0.686i)53-s + 1.53·61-s + (−0.585 − 0.585i)73-s − 81-s − 1.69i·89-s + (−0.507 + 0.507i)97-s − 0.199·101-s − 0.574i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.433931709\)
\(L(\frac12)\) \(\approx\) \(1.433931709\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (5 + 5i)T + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−9.500237568668142842893983551188, −8.287930654625881369253654658357, −7.61040917803321066918598047893, −7.21314894823549302892280489246, −5.72123793238406465915987156058, −5.28335749916440827837260673826, −4.37204691372230035009015211978, −3.01561917309734725438421389998, −2.33298662812018307122173189170, −0.61260532860234294807589608905, 1.22659962426848624835611065649, 2.50266323062434627352120824100, 3.66124572424794132344560121879, 4.42967311077520984884769525126, 5.49081465104499543685072245073, 6.38341562420334819823389927008, 7.07564203753473218927458952273, 7.918244794743921448562385195935, 8.851430931675751862171392110095, 9.601592281465952282014532857786

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