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Properties

Label 2-800-8.3-c0-0-1
Degree $2$
Conductor $800$
Sign $1$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 11-s − 17-s + 19-s − 27-s + 33-s − 41-s − 2·43-s + 49-s − 51-s + 57-s − 2·59-s + 67-s − 73-s − 81-s + 83-s − 89-s + 2·97-s + 107-s − 113-s + ⋯
L(s)  = 1  + 3-s + 11-s − 17-s + 19-s − 27-s + 33-s − 41-s − 2·43-s + 49-s − 51-s + 57-s − 2·59-s + 67-s − 73-s − 81-s + 83-s − 89-s + 2·97-s + 107-s − 113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800} (751, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311746264\)
\(L(\frac12)\) \(\approx\) \(1.311746264\)
\(L(1)\) 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 \( 1 \)
good3 \( 1 - T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 + T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 - T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( ( 1 - T )^{2} \)
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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

−10.31384706848473180457868494355, −9.369370861169465024301083910326, −8.868077097882977508829880360178, −8.064200851859634343342378962440, −7.11087489607727481549774192931, −6.23923712096805878863340366520, −5.01325669588277177462905749820, −3.86101044046500691129164586929, −3.00869154103659649832886212316, −1.76060560640469529675214210156, 1.76060560640469529675214210156, 3.00869154103659649832886212316, 3.86101044046500691129164586929, 5.01325669588277177462905749820, 6.23923712096805878863340366520, 7.11087489607727481549774192931, 8.064200851859634343342378962440, 8.868077097882977508829880360178, 9.369370861169465024301083910326, 10.31384706848473180457868494355

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