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Properties

Label 2-40e2-20.3-c1-0-4
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 + (−1 − i)13-s + (−3 + 3i)17-s + 4i·29-s + (−7 + 7i)37-s − 8·41-s − 7i·49-s + (9 + 9i)53-s − 12·61-s + (11 + 11i)73-s − 9·81-s + 16i·89-s + (−13 + 13i)97-s − 2·101-s − 6i·109-s + ⋯
L(s)  = 1  + i·9-s + (−0.277 − 0.277i)13-s + (−0.727 + 0.727i)17-s + 0.742i·29-s + (−1.15 + 1.15i)37-s − 1.24·41-s − i·49-s + (1.23 + 1.23i)53-s − 1.53·61-s + (1.28 + 1.28i)73-s − 81-s + 1.69i·89-s + (−1.31 + 1.31i)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

\(L(1)\) \(\approx\) \(0.9589279285\)
\(L(\frac12)\) \(\approx\) \(0.9589279285\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (7 - 7i)T - 37iT^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11 - 11i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−9.767903981935925295708492327207, −8.687258232474921851433377090122, −8.226234801848473967582035547826, −7.25980775975196894870560459596, −6.55043374149371581466503007737, −5.44955610623167730756439373762, −4.81645057521060485154422430691, −3.76485928125883358608548295484, −2.62941979123255925439395993849, −1.58776063976500662580433775881, 0.35694884930754046654729483871, 1.92297272797417177597877392857, 3.10195229467936784608381047703, 4.04574733067508784466632777220, 4.95607936423897530364484505212, 5.95840057996330046255882319759, 6.77304054145758788074451628387, 7.39847823269285495455918527513, 8.505857169049932365334168389233, 9.158047673976675898491476918859

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