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Properties

Label 2-800-20.3-c1-0-7
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 + (1 + i)7-s + i·9-s + 6i·11-s + (1 + i)13-s + (−1 + i)17-s + 4·19-s + 2·21-s + (−5 + 5i)23-s + (4 + 4i)27-s − 8i·29-s − 2i·31-s + (6 + 6i)33-s + (5 − 5i)37-s + 2·39-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (0.377 + 0.377i)7-s + 0.333i·9-s + 1.80i·11-s + (0.277 + 0.277i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s + 0.436·21-s + (−1.04 + 1.04i)23-s + (0.769 + 0.769i)27-s − 1.48i·29-s − 0.359i·31-s + (1.04 + 1.04i)33-s + (0.821 − 0.821i)37-s + 0.320·39-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: $\chi_{800} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82831 + 0.519387i\)
\(L(\frac12)\) \(\approx\) \(1.82831 + 0.519387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (5 - 5i)T - 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (7 + 7i)T + 47iT^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-7 - 7i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (9 + 9i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
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

−10.06482760269775380371040344600, −9.578548788194541063102053656936, −8.493892987237767762568860655333, −7.68095093311166738795982055461, −7.21360138864090689770640421362, −5.98755994956523760813513073468, −4.92520628464722690442023488985, −3.94881765763319959652114720674, −2.39316962120136038256601805581, −1.73618760263876091414599865268, 0.956690192267222933516784088180, 2.90160514683644799411765705493, 3.58608957262901790098083189385, 4.64257544361796656521375668821, 5.78637342642598754756864453424, 6.64114012365709273266181391516, 7.930608750462495142989773829040, 8.525634426571618892253276351088, 9.253414012269336771168943351682, 10.18656293180407804554371281890

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