↑

Properties

Label 2-20e2-80.27-c1-0-21
Degree $2$
Conductor $400$
Sign $0.574 - 0.818i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
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  + (−0.430 + 1.34i)2-s + 2.96·3-s + (−1.62 − 1.15i)4-s + (−1.27 + 3.99i)6-s + (0.115 + 0.115i)7-s + (2.26 − 1.69i)8-s + 5.79·9-s + (2.95 − 2.95i)11-s + (−4.83 − 3.43i)12-s + 1.55i·13-s + (−0.204 + 0.105i)14-s + (1.31 + 3.77i)16-s + (−0.299 − 0.299i)17-s + (−2.49 + 7.80i)18-s + (−2.26 + 2.26i)19-s + ⋯
L(s)  = 1  + (−0.304 + 0.952i)2-s + 1.71·3-s + (−0.814 − 0.579i)4-s + (−0.520 + 1.63i)6-s + (0.0435 + 0.0435i)7-s + (0.800 − 0.599i)8-s + 1.93·9-s + (0.892 − 0.892i)11-s + (−1.39 − 0.992i)12-s + 0.432i·13-s + (−0.0546 + 0.0282i)14-s + (0.327 + 0.944i)16-s + (−0.0726 − 0.0726i)17-s + (−0.587 + 1.84i)18-s + (−0.519 + 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72871 + 0.898459i\)
\(L(\frac12)\) \(\approx\) \(1.72871 + 0.898459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.430 - 1.34i)T \)
5 \( 1 \)
good3 \( 1 - 2.96T + 3T^{2} \)
7 \( 1 + (-0.115 - 0.115i)T + 7iT^{2} \)
11 \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
17 \( 1 + (0.299 + 0.299i)T + 17iT^{2} \)
19 \( 1 + (2.26 - 2.26i)T - 19iT^{2} \)
23 \( 1 + (4.14 - 4.14i)T - 23iT^{2} \)
29 \( 1 + (-0.289 - 0.289i)T + 29iT^{2} \)
31 \( 1 + 4.18iT - 31T^{2} \)
37 \( 1 + 1.63iT - 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 + 6.72iT - 43T^{2} \)
47 \( 1 + (4.38 - 4.38i)T - 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (-1.63 - 1.63i)T + 59iT^{2} \)
61 \( 1 + (1.23 - 1.23i)T - 61iT^{2} \)
67 \( 1 + 2.49iT - 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 - 1.12i)T + 73iT^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 + 1.62T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (9.69 + 9.69i)T + 97iT^{2} \)
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

−11.27092629434881597658698126951, −9.888636059220072258627773310622, −9.358152018409894224281443124179, −8.482869196683352108527824097752, −7.988572128981970757300235210355, −6.93988460767348794668970276500, −5.93434041545562781312645671431, −4.31903706364106337373720842847, −3.47097836470046656693291069940, −1.72575939926283973615588110216, 1.69777510763291602364092938825, 2.71446289583778712292463697796, 3.81194167935183579872674653335, 4.64290411364883356772533851443, 6.79049328840012410717351461083, 7.87999612855121979395000781223, 8.546937799118248234005109940859, 9.363272344790414915446184779953, 9.969726505543335820806654751671, 10.94678514079389864296328966023

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