Properties

Label 2-40e2-5.4-c1-0-30
Degree $2$
Conductor $1600$
Sign $-0.894 + 0.447i$
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  i·3-s − 2i·7-s + 2·9-s − 3·11-s − 4i·13-s − 3i·17-s − 5·19-s − 2·21-s + 6i·23-s − 5i·27-s − 2·31-s + 3i·33-s − 2i·37-s − 4·39-s − 3·41-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.755i·7-s + 0.666·9-s − 0.904·11-s − 1.10i·13-s − 0.727i·17-s − 1.14·19-s − 0.436·21-s + 1.25i·23-s − 0.962i·27-s − 0.359·31-s + 0.522i·33-s − 0.328i·37-s − 0.640·39-s − 0.468·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069747472\)
\(L(\frac12)\) \(\approx\) \(1.069747472\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
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

−9.033129279835063836287112328697, −7.979946653368058210238291303002, −7.52265525667723835458661791976, −6.85912414369834048784166163008, −5.81944050046625790123303105400, −4.95642289359197992827615876786, −3.96626328181030401313763663332, −2.90176831911550508098756468628, −1.71065455482548057711580589783, −0.39734728218883045740931514747, 1.77825955755040677392653954616, 2.73991896352042329987197508111, 4.08701952912088618592638500784, 4.60083420206572865839383475998, 5.63412177363564889887943413343, 6.48520933267338993531911959906, 7.30643623631575389702948724994, 8.454802171153744946770242249215, 8.839874986700037821240829008085, 9.833473105060152715516327423133

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