Properties

Label 2-800-5.4-c1-0-11
Degree $2$
Conductor $800$
Sign $0.894 + 0.447i$
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  + 3·9-s − 6i·13-s + 2i·17-s + 10·29-s − 2i·37-s + 10·41-s + 7·49-s − 14i·53-s − 10·61-s + 6i·73-s + 9·81-s − 10·89-s + 18i·97-s − 2·101-s − 6·109-s + ⋯
L(s)  = 1  + 9-s − 1.66i·13-s + 0.485i·17-s + 1.85·29-s − 0.328i·37-s + 1.56·41-s + 49-s − 1.92i·53-s − 1.28·61-s + 0.702i·73-s + 81-s − 1.05·89-s + 1.82i·97-s − 0.199·101-s − 0.574·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61397 - 0.381007i\)
\(L(\frac12)\) \(\approx\) \(1.61397 - 0.381007i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−10.33520937171866579085045591813, −9.480292352571468861477527396633, −8.364033683732178246378748509922, −7.72030003281674611875320911196, −6.75814582302984525899496199797, −5.78985949289723735257272644440, −4.80272843830423347394549676359, −3.76504169895643042056289074653, −2.58961987590985827320755071125, −1.00561549055021288502486662352, 1.34455815818838023794816884635, 2.66520218597480113038850523565, 4.16368315818889048506434704127, 4.68213637693609946801652378695, 6.09645881279135912507132147456, 6.90485813285815305524578174386, 7.61372460765992789645661953803, 8.802311016877131148333845890323, 9.451278719798306226364636225712, 10.27707991207924662442301518911

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