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Properties

Label 2-800-32.5-c1-0-70
Degree $2$
Conductor $800$
Sign $-0.968 - 0.248i$
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  + (1.37 − 0.315i)2-s + (−1.29 − 3.12i)3-s + (1.80 − 0.868i)4-s + (−2.77 − 3.90i)6-s + (−1.02 − 1.02i)7-s + (2.20 − 1.76i)8-s + (−5.97 + 5.97i)9-s + (0.473 − 1.14i)11-s + (−5.04 − 4.50i)12-s + (−5.17 + 2.14i)13-s + (−1.73 − 1.09i)14-s + (2.49 − 3.13i)16-s − 4.22i·17-s + (−6.35 + 10.1i)18-s + (−3.63 + 1.50i)19-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (−0.747 − 1.80i)3-s + (0.900 − 0.434i)4-s + (−1.13 − 1.59i)6-s + (−0.387 − 0.387i)7-s + (0.781 − 0.624i)8-s + (−1.99 + 1.99i)9-s + (0.142 − 0.344i)11-s + (−1.45 − 1.30i)12-s + (−1.43 + 0.594i)13-s + (−0.464 − 0.291i)14-s + (0.622 − 0.782i)16-s − 1.02i·17-s + (−1.49 + 2.38i)18-s + (−0.833 + 0.345i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194729 + 1.54423i\)
\(L(\frac12)\) \(\approx\) \(0.194729 + 1.54423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.315i)T \)
5 \( 1 \)
good3 \( 1 + (1.29 + 3.12i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.02 + 1.02i)T + 7iT^{2} \)
11 \( 1 + (-0.473 + 1.14i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.17 - 2.14i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 4.22iT - 17T^{2} \)
19 \( 1 + (3.63 - 1.50i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.53 + 5.53i)T - 23iT^{2} \)
29 \( 1 + (-0.848 - 2.04i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 0.964T + 31T^{2} \)
37 \( 1 + (2.41 + 0.998i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.62 + 4.62i)T - 41iT^{2} \)
43 \( 1 + (0.394 - 0.951i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.36iT - 47T^{2} \)
53 \( 1 + (1.51 - 3.65i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.648 + 0.268i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.01 - 7.28i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.684 + 1.65i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.24 - 6.24i)T + 71iT^{2} \)
73 \( 1 + (-5.70 + 5.70i)T - 73iT^{2} \)
79 \( 1 + 14.9iT - 79T^{2} \)
83 \( 1 + (3.35 - 1.38i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (9.90 + 9.90i)T + 89iT^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−10.21449030760533000555750130104, −8.791855098192937110340728418012, −7.48571247385360643638336299953, −7.00682346883469180483405444164, −6.43522010337190406024101011610, −5.43269174110930364038387975149, −4.60844199939547801766487351543, −2.88431024422706240754804053676, −2.03122396776441383377222943757, −0.58024382884358756978907624218, 2.69854644358892251353566681060, 3.65010457829855418383949298185, 4.57275358339308166435455168365, 5.22347810285833743521055394342, 5.96711367375167689283802500523, 6.88019490980992643092350426130, 8.217151659777977982348346052839, 9.365680699756530562654517157781, 9.975753449714215200630326534195, 10.86133387135669973566880214871

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