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Properties

Label 2-800-25.2-c0-0-0
Degree $2$
Conductor $800$
Sign $0.940 + 0.338i$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)5-s + (0.587 − 0.809i)9-s + (−0.896 + 0.142i)13-s + (−0.142 + 0.278i)17-s + (0.809 − 0.587i)25-s + (−1.11 − 0.363i)29-s + (0.309 + 1.95i)37-s + (1.53 + 1.11i)41-s + (0.309 − 0.951i)45-s − i·49-s + (−0.809 − 1.58i)53-s + (−1.53 + 1.11i)61-s + (−0.809 + 0.412i)65-s + (−0.278 + 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)5-s + (0.587 − 0.809i)9-s + (−0.896 + 0.142i)13-s + (−0.142 + 0.278i)17-s + (0.809 − 0.587i)25-s + (−1.11 − 0.363i)29-s + (0.309 + 1.95i)37-s + (1.53 + 1.11i)41-s + (0.309 − 0.951i)45-s − i·49-s + (−0.809 − 1.58i)53-s + (−1.53 + 1.11i)61-s + (−0.809 + 0.412i)65-s + (−0.278 + 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.940 + 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.128863154\)
\(L(\frac12)\) \(\approx\) \(1.128863154\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
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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

−10.08045400513623570008867906433, −9.707911548797765802535774213227, −8.936496998061031364830879666774, −7.86782346630355366693036084241, −6.82086715214551993267658918981, −6.11815406251195711050683927561, −5.08036235295545432027713485584, −4.14287124484949957402032370697, −2.75790099364208801231292043780, −1.47999659328552296763977641432, 1.83319920049817822845134952166, 2.76521078463798749230212247285, 4.27553742926656815275532791370, 5.26504644967036347273775515883, 6.04439033722852082247240314522, 7.25571705213421627802553292887, 7.67006156587674893769099588919, 9.190040901410567931318648116818, 9.527561730856915754922740394580, 10.72948425127614984686947302403

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