↑

Properties

Label 2-800-25.14-c1-0-20
Degree $2$
Conductor $800$
Sign $-0.202 + 0.979i$
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  + (−1.59 + 1.56i)5-s + (−2.42 − 1.76i)9-s + (4.08 − 5.61i)13-s + (−7.27 + 2.36i)17-s + (0.0759 − 4.99i)25-s + (3.22 − 9.93i)29-s + (6.39 − 8.80i)37-s + (−3.65 − 2.65i)41-s + (6.63 − 0.999i)45-s + 7·49-s + (−13.8 − 4.49i)53-s + (−11.7 + 8.52i)61-s + (2.31 + 15.3i)65-s + (−6.26 − 8.61i)73-s + (2.78 + 8.55i)81-s + ⋯
L(s)  = 1  + (−0.712 + 0.701i)5-s + (−0.809 − 0.587i)9-s + (1.13 − 1.55i)13-s + (−1.76 + 0.573i)17-s + (0.0151 − 0.999i)25-s + (0.599 − 1.84i)29-s + (1.05 − 1.44i)37-s + (−0.570 − 0.414i)41-s + (0.988 − 0.148i)45-s + 49-s + (−1.90 − 0.617i)53-s + (−1.50 + 1.09i)61-s + (0.286 + 1.90i)65-s + (−0.732 − 1.00i)73-s + (0.309 + 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.202 + 0.979i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478881 - 0.587899i\)
\(L(\frac12)\) \(\approx\) \(0.478881 - 0.587899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 1.56i)T \)
good3 \( 1 + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.08 + 5.61i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (7.27 - 2.36i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-3.22 + 9.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.39 + 8.80i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.65 + 2.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (13.8 + 4.49i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (11.7 - 8.52i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.26 + 8.61i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.06 - 0.772i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (16.2 + 5.26i)T + (78.4 + 57.0i)T^{2} \)
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

−10.24446092944310525432673458606, −8.988784428943217416825115115875, −8.318191314162715170027914508372, −7.57373854758510669418740078735, −6.32400290703700195788044069559, −5.94109862060267591912874284527, −4.37184896867683532014575739497, −3.48567523922429645785829725528, −2.52377504488643121583277262541, −0.37867870932806123214902618622, 1.57008115108962011758952856566, 3.05144898054607576407293320616, 4.35515635580386379667878340472, 4.88937189305254533006841788265, 6.23510124406555851464614047353, 7.00721929602441708003689791332, 8.191076324658007521327327347980, 8.787830105572980079705037176079, 9.319997922980854928909312124200, 10.85346789106630230788864996539

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