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Properties

Label 2-40e2-40.29-c1-0-15
Degree $2$
Conductor $1600$
Sign $0.200 - 0.979i$
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  + 2.73·3-s + 4.73i·7-s + 4.46·9-s + 3.46i·11-s − 3.46·13-s − 3.46i·17-s + 2i·19-s + 12.9i·21-s + 2.19i·23-s + 3.99·27-s − 2.53·31-s + 9.46i·33-s + 6·37-s − 9.46·39-s + 9.46·41-s + ⋯
L(s)  = 1  + 1.57·3-s + 1.78i·7-s + 1.48·9-s + 1.04i·11-s − 0.960·13-s − 0.840i·17-s + 0.458i·19-s + 2.82i·21-s + 0.457i·23-s + 0.769·27-s − 0.455·31-s + 1.64i·33-s + 0.986·37-s − 1.51·39-s + 1.47·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.200 - 0.979i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.719357829\)
\(L(\frac12)\) \(\approx\) \(2.719357829\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
7 \( 1 - 4.73iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 2.19iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 + 0.196T + 43T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 0.928iT - 61T^{2} \)
67 \( 1 - 0.196T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + 6.39iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 1.26T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−9.500621962928108249037186223300, −8.966384779284115140496025792257, −7.923419170602528345343824394900, −7.59970608771549682618107341557, −6.45903669011456216876794986536, −5.34466375302736938151151565560, −4.55365316953575248637947971889, −3.30697522565437927033711087196, −2.46102723005948952091410348880, −1.98417071091467643830613287298, 0.847790244608192170227646061092, 2.24132143991853128225048977259, 3.27641404066480984519586420917, 3.93370253131450462495579652044, 4.72489100613936883255641546037, 6.20429962429969083927797120553, 7.15597457671321222605002591015, 7.76153614483563570440183686940, 8.301976176704695276385202007473, 9.239417469734313375508888528379

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