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Properties

Label 2-40e2-5.4-c1-0-17
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

\(L(1)\) \(\approx\) \(1.996966738\)
\(L(\frac12)\) \(\approx\) \(1.996966738\)
\(L(\frac{3}{2})\) 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

−9.537298808825653968456963022092, −8.466732275729835641972245163985, −7.59429110607486304330216569176, −6.92206168735307911302235535747, −6.42963706496312371619093103027, −5.22013194175450092371537849598, −4.14354412034294634340637342580, −3.54741066566987494914322703675, −1.88649838030091600330961784947, −1.12105277475785692415343021852, 1.04352949765364317172325986919, 2.56412917424182407302898250736, 3.48975293134307007781769081762, 4.51635204004326823266481540614, 5.26917074365722008247656327573, 6.14190369361402019347132263409, 7.09069166000098794953437238075, 7.896582581441935673004699124551, 8.907364742296679397927542322911, 9.413122181501121067502585812993

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