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Properties

Label 2-800-100.71-c0-0-0
Degree $2$
Conductor $800$
Sign $-0.0941 - 0.995i$
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  + (−0.587 + 0.809i)3-s + (0.309 + 0.951i)5-s − 0.618i·7-s + (0.190 + 0.587i)13-s + (−0.951 − 0.309i)15-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (0.587 + 0.809i)31-s + (0.587 − 0.190i)35-s + (−0.309 − 0.951i)37-s + (−0.587 − 0.190i)39-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s + (0.309 + 0.951i)5-s − 0.618i·7-s + (0.190 + 0.587i)13-s + (−0.951 − 0.309i)15-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (−0.951 − 0.309i)23-s + (−0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (−1.30 − 0.951i)29-s + (0.587 + 0.809i)31-s + (0.587 − 0.190i)35-s + (−0.309 − 0.951i)37-s + (−0.587 − 0.190i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.0941 - 0.995i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ -0.0941 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8251445280\)
\(L(\frac12)\) \(\approx\) \(0.8251445280\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.52756790468046349398885839451, −10.07809852863642696903753293528, −9.389826760360603742359799172187, −7.971448270479543526254220546799, −7.24915505465419812746129424405, −6.17546387219881210045281667447, −5.49524623658445323878659103246, −4.24642297469687367469854570734, −3.55245831630499198999208734966, −1.99189397916901956426965739668, 0.981773254448728305807019379431, 2.30666435097973230087939109912, 3.86561421433811943132965035200, 5.33371631426787569988861624613, 5.62309906939299373268571904652, 6.74925336414501811004048243289, 7.60343063309179130158448965326, 8.614397782725758398348348424658, 9.300489623427658675355847937371, 10.16088377767830453520734615054

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