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Properties

Label 2-800-100.11-c0-0-0
Degree $2$
Conductor $800$
Sign $0.612 - 0.790i$
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)3-s + (−0.809 − 0.587i)5-s + 1.61i·7-s + (1.30 + 0.951i)13-s + (0.587 + 0.809i)15-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (0.951 − 1.30i)35-s + (0.809 + 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + (−0.809 − 0.587i)5-s + 1.61i·7-s + (1.30 + 0.951i)13-s + (0.587 + 0.809i)15-s + (−0.587 + 0.190i)19-s + (0.500 − 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (−0.190 + 0.587i)29-s + (0.951 − 0.309i)31-s + (0.951 − 1.30i)35-s + (0.809 + 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5641007637\)
\(L(\frac12)\) \(\approx\) \(0.5641007637\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−11.13235746859653710027954389613, −9.568961502346711834596502097167, −8.717981016387441683898159566088, −8.333387192106487678352513901161, −6.93563737475737013129668415588, −6.09735624353967895357515333516, −5.44528026384403134791621532595, −4.43959128520167464188661590950, −3.16359254788662527190539733778, −1.50269268871359637148224907544, 0.72410527832944070341814738241, 3.07466700820151543754262244854, 4.07085834810941884673309395035, 4.79450702754302185612680932296, 6.16339941026947681820757163648, 6.70994498706297363980216116141, 7.82178584472071459942098804806, 8.359431168317510759850937031981, 9.916495128276396188435742587749, 10.76421290348671592221019667730

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