Properties

Label 2-800-4.3-c2-0-17
Degree $2$
Conductor $800$
Sign $0.707 - 0.707i$
Analytic cond. $21.7984$
Root an. cond. $4.66887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.75i·3-s − 3.84i·7-s + 1.41·9-s + 6.19i·11-s + 16.1·13-s − 5.20·17-s − 36.2i·19-s + 10.6·21-s + 22.0i·23-s + 28.6i·27-s + 20.0·29-s − 26.4i·31-s − 17.0·33-s + 69.3·37-s + 44.3i·39-s + ⋯
L(s)  = 1  + 0.918i·3-s − 0.549i·7-s + 0.157·9-s + 0.562i·11-s + 1.23·13-s − 0.306·17-s − 1.90i·19-s + 0.504·21-s + 0.958i·23-s + 1.06i·27-s + 0.690·29-s − 0.852i·31-s − 0.516·33-s + 1.87·37-s + 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(21.7984\)
Root analytic conductor: \(4.66887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.069642407\)
\(L(\frac12)\) \(\approx\) \(2.069642407\)
\(L(2)\) 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 \( 1 - 2.75iT - 9T^{2} \)
7 \( 1 + 3.84iT - 49T^{2} \)
11 \( 1 - 6.19iT - 121T^{2} \)
13 \( 1 - 16.1T + 169T^{2} \)
17 \( 1 + 5.20T + 289T^{2} \)
19 \( 1 + 36.2iT - 361T^{2} \)
23 \( 1 - 22.0iT - 529T^{2} \)
29 \( 1 - 20.0T + 841T^{2} \)
31 \( 1 + 26.4iT - 961T^{2} \)
37 \( 1 - 69.3T + 1.36e3T^{2} \)
41 \( 1 - 11.6T + 1.68e3T^{2} \)
43 \( 1 + 25.8iT - 1.84e3T^{2} \)
47 \( 1 - 66.1iT - 2.20e3T^{2} \)
53 \( 1 - 39.5T + 2.80e3T^{2} \)
59 \( 1 + 27.7iT - 3.48e3T^{2} \)
61 \( 1 + 54.1T + 3.72e3T^{2} \)
67 \( 1 - 107. iT - 4.48e3T^{2} \)
71 \( 1 - 70.7iT - 5.04e3T^{2} \)
73 \( 1 + 37.4T + 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 - 6.40T + 9.40e3T^{2} \)
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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

−10.08520175886183248985261589869, −9.446910645135695683066234755207, −8.702510708851886544724216027752, −7.53315944194217920217168361584, −6.76804226275593960312021754749, −5.64128532399189099088698549714, −4.47824046182685693043400685853, −4.03952535439498710256315709127, −2.72951988459881387882698495541, −1.04665058182290120798737263806, 0.952716885536687175366369809747, 2.03413222821155893138646672702, 3.34255171861631631915988091512, 4.48714837638434181283359026593, 6.03922478618646135086809283908, 6.19420573947752110942148661984, 7.44242051297869911771879625975, 8.282063031483948974307607091846, 8.819558499625834289308724479410, 10.06617316948082995267546934919

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