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Properties

Label 2-800-100.23-c1-0-10
Degree $2$
Conductor $800$
Sign $0.909 + 0.415i$
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  + (−0.440 − 0.864i)3-s + (−2.21 − 0.268i)5-s + (2.34 + 2.34i)7-s + (1.21 − 1.66i)9-s + (−1.76 − 2.43i)11-s + (0.910 + 5.74i)13-s + (0.745 + 2.03i)15-s + (3.39 + 1.72i)17-s + (−0.0821 − 0.252i)19-s + (0.995 − 3.06i)21-s + (1.20 − 7.61i)23-s + (4.85 + 1.19i)25-s + (−4.84 − 0.767i)27-s + (3.49 + 1.13i)29-s + (2.21 − 0.718i)31-s + ⋯
L(s)  = 1  + (−0.254 − 0.498i)3-s + (−0.992 − 0.119i)5-s + (0.888 + 0.888i)7-s + (0.403 − 0.555i)9-s + (−0.532 − 0.732i)11-s + (0.252 + 1.59i)13-s + (0.192 + 0.525i)15-s + (0.822 + 0.419i)17-s + (−0.0188 − 0.0579i)19-s + (0.217 − 0.668i)21-s + (0.251 − 1.58i)23-s + (0.971 + 0.238i)25-s + (−0.932 − 0.147i)27-s + (0.649 + 0.211i)29-s + (0.397 − 0.129i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.909 + 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31750 - 0.287006i\)
\(L(\frac12)\) \(\approx\) \(1.31750 - 0.287006i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.268i)T \)
good3 \( 1 + (0.440 + 0.864i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-2.34 - 2.34i)T + 7iT^{2} \)
11 \( 1 + (1.76 + 2.43i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.910 - 5.74i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.39 - 1.72i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.0821 + 0.252i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.20 + 7.61i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-3.49 - 1.13i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.21 + 0.718i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-10.1 + 1.61i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (1.02 + 0.746i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.20 + 2.20i)T - 43iT^{2} \)
47 \( 1 + (-4.22 + 2.15i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.44 + 2.77i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-9.55 - 6.94i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.05 - 3.67i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-6.95 + 13.6i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (8.86 + 2.87i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.17 + 0.820i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (3.72 - 11.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.11 + 3.62i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-4.48 - 6.17i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.61 + 3.16i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.38092326828870873258844256642, −9.036394669860614096708635165057, −8.506653637534066598736415645390, −7.70661233154138736418694857027, −6.74489421024521462253830750296, −5.90727568832444002954147550623, −4.73831691114604171210347264855, −3.89744172836663170132618392858, −2.44870113841502753368054506514, −0.982255376199959325779511627012, 1.06019859087962740652880594416, 2.95447390849644599248784568797, 4.10089394978385143547729132057, 4.81742296729287772079167079117, 5.62568683611614752784753706479, 7.41005233398253852146098027443, 7.57376299413759587818052235461, 8.315992898759100417320838089920, 9.875302470007042177672063871330, 10.31800861529461839019110857250

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