Properties

Label 2-800-40.29-c1-0-11
Degree $2$
Conductor $800$
Sign $-0.663 + 0.748i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.732·3-s + 2.73i·7-s − 2.46·9-s − 2i·11-s − 3.46·13-s − 3.46i·17-s − 7.46i·19-s − 2i·21-s + 4.19i·23-s + 4·27-s − 6.92i·29-s − 1.46·31-s + 1.46i·33-s − 2·37-s + 2.53·39-s + ⋯
L(s)  = 1  − 0.422·3-s + 1.03i·7-s − 0.821·9-s − 0.603i·11-s − 0.960·13-s − 0.840i·17-s − 1.71i·19-s − 0.436i·21-s + 0.874i·23-s + 0.769·27-s − 1.28i·29-s − 0.262·31-s + 0.254i·33-s − 0.328·37-s + 0.406·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.663 + 0.748i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.168011 - 0.373536i\)
\(L(\frac12)\) \(\approx\) \(0.168011 - 0.373536i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 + 7.46iT - 19T^{2} \)
23 \( 1 - 4.19iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 + 8.73T + 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 + 4.53T + 53T^{2} \)
59 \( 1 - 0.535iT - 59T^{2} \)
61 \( 1 + 4.92iT - 61T^{2} \)
67 \( 1 + 7.26T + 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 + 0.535iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 - 6.39iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.843151173190354857571567810624, −9.092544269166637624577617759526, −8.413916345956317663786644901892, −7.30400507093415134794035337227, −6.35806061421585888655390720022, −5.40453170544000197434787236152, −4.88407789672504954709828188624, −3.18200286231576992832809567057, −2.34239977120619844696625547455, −0.20530186332446194564206081423, 1.64427633098665513122087339943, 3.19875189008048115435963113536, 4.29686576740692511376344309883, 5.23137781273671308294036010667, 6.24051649265332756439272833205, 7.10814942907766504385392273905, 7.959631945086510105715149545360, 8.815802603217801937837244232209, 10.14278536454825134496276268351, 10.33971338155455910654517772829

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