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Properties

Label 2-800-160.29-c1-0-4
Degree $2$
Conductor $800$
Sign $-0.264 - 0.964i$
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  − 1.41·2-s + (0.292 − 0.707i)3-s + 2.00·4-s + (−0.414 + 1.00i)6-s + (−1 − i)7-s − 2.82·8-s + (1.70 + 1.70i)9-s + (−4.12 + 1.70i)11-s + (0.585 − 1.41i)12-s + (0.707 + 0.292i)13-s + (1.41 + 1.41i)14-s + 4.00·16-s − 2.82·17-s + (−2.41 − 2.41i)18-s + (−1.53 + 3.70i)19-s + ⋯
L(s)  = 1  − 1.00·2-s + (0.169 − 0.408i)3-s + 1.00·4-s + (−0.169 + 0.408i)6-s + (−0.377 − 0.377i)7-s − 1.00·8-s + (0.569 + 0.569i)9-s + (−1.24 + 0.514i)11-s + (0.169 − 0.408i)12-s + (0.196 + 0.0812i)13-s + (0.377 + 0.377i)14-s + 1.00·16-s − 0.685·17-s + (−0.569 − 0.569i)18-s + (−0.352 + 0.850i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.264 - 0.964i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300372 + 0.393689i\)
\(L(\frac12)\) \(\approx\) \(0.300372 + 0.393689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 \)
good3 \( 1 + (-0.292 + 0.707i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (4.12 - 1.70i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.707 - 0.292i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (1.53 - 3.70i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.82 - 5.82i)T - 23iT^{2} \)
29 \( 1 + (-3.12 - 1.29i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.171 + 0.171i)T + 41iT^{2} \)
43 \( 1 + (-1.94 - 4.70i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 + (0.464 + 1.12i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.87 - 4.53i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.29 - 5.53i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (5.82 - 5.82i)T - 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-4.53 - 1.87i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.65 - 8.65i)T - 89iT^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−10.23202449528581467297999658553, −9.907575820298720015438336202501, −8.709496014269729955582586102010, −7.84453431084287399471193665931, −7.39209122095988472753278432523, −6.46940220170287833789781715992, −5.40326245755909615426051525736, −3.98257246566848786419816149631, −2.56330722072778295793989448181, −1.60413881787714813896679488978, 0.31559933256181672543311472512, 2.23903773794713734637222249752, 3.19268788643433014718892185210, 4.52692022777735404628327821111, 5.88745484842076707227250454662, 6.60955456609439166783867240541, 7.61710249737148169314875060431, 8.573723384227398633107604219688, 9.071834702998275022779416541806, 10.07225409748685699797420133923

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