Properties

Label 2-40e2-40.29-c1-0-6
Degree $2$
Conductor $1600$
Sign $0.663 - 0.748i$
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  − 3-s − 3.46i·7-s − 2·9-s + 3i·11-s − 3.46·13-s + 3i·17-s i·19-s + 3.46i·21-s + 5·27-s + 10.3i·29-s + 6.92·31-s − 3i·33-s + 10.3·37-s + 3.46·39-s − 9·41-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.30i·7-s − 0.666·9-s + 0.904i·11-s − 0.960·13-s + 0.727i·17-s − 0.229i·19-s + 0.755i·21-s + 0.962·27-s + 1.92i·29-s + 1.24·31-s − 0.522i·33-s + 1.70·37-s + 0.554·39-s − 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9469744145\)
\(L(\frac12)\) \(\approx\) \(0.9469744145\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 3T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.723001314476906966876808188399, −8.691257436223008386790857807810, −7.79174747662742675318894123035, −7.02482506751260008589971485695, −6.45315831269802301340573313322, −5.25729243646382768443004688434, −4.64422844565878597205287770010, −3.67396254512510274084575324030, −2.44521571234295434670274470986, −0.982186635103503665799593854644, 0.48810051878648358844757487152, 2.39152664843649876518415424324, 2.98292513368384897572339072330, 4.48957355605712433354969311397, 5.36111100221311183335086572718, 5.95509094023615781303372443677, 6.61225455370843592692080664224, 7.967538866104044693940380151143, 8.375996117283724918355967058943, 9.438507946227640046369424483948

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