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Properties

Label 2-800-20.7-c1-0-1
Degree $2$
Conductor $800$
Sign $-0.850 - 0.525i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
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  + (1 + i)3-s + (−3 + 3i)7-s − i·9-s + 2i·11-s + (−3 + 3i)13-s + (−1 − i)17-s − 4·19-s − 6·21-s + (−1 − i)23-s + (4 − 4i)27-s + 10i·31-s + (−2 + 2i)33-s + (1 + i)37-s − 6·39-s − 10·41-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−1.13 + 1.13i)7-s − 0.333i·9-s + 0.603i·11-s + (−0.832 + 0.832i)13-s + (−0.242 − 0.242i)17-s − 0.917·19-s − 1.30·21-s + (−0.208 − 0.208i)23-s + (0.769 − 0.769i)27-s + 1.79i·31-s + (−0.348 + 0.348i)33-s + (0.164 + 0.164i)37-s − 0.960·39-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272551 + 0.959422i\)
\(L(\frac12)\) \(\approx\) \(0.272551 + 0.959422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (3 - 3i)T - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-3 - 3i)T + 97iT^{2} \)
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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

−10.19919946969799408617834836787, −9.724690470722904152278049104281, −8.991661113158697161383666627446, −8.452724393400733958631181553980, −6.92574740119835516732910322817, −6.44874293123435635520929336145, −5.17525279931901536021265829349, −4.18371058760003549663218260902, −3.09577955010136171888452269471, −2.22325946034601879952506452022, 0.42870044250894246003153245453, 2.21567402034006199968334476966, 3.26785303674686534517367493068, 4.24383002231607724826439919838, 5.60819060416649640444834770186, 6.63422051205586832692203262238, 7.38569260731394457793209783807, 8.069463989498949098821476824004, 8.987052401958675011738244603072, 10.12019770827379467208749869707

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