↑

Properties

Label 2-20e2-20.7-c1-0-4
Degree $2$
Conductor $400$
Sign $0.525 - 0.850i$
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  + (2.23 + 2.23i)3-s + (2.23 − 2.23i)7-s + 7.00i·9-s + 10.0·21-s + (−6.70 − 6.70i)23-s + (−8.94 + 8.94i)27-s + 6i·29-s − 12·41-s + (−2.23 − 2.23i)43-s + (6.70 − 6.70i)47-s − 3.00i·49-s + 8·61-s + (15.6 + 15.6i)63-s + (11.1 − 11.1i)67-s − 30.0i·69-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)3-s + (0.845 − 0.845i)7-s + 2.33i·9-s + 2.18·21-s + (−1.39 − 1.39i)23-s + (−1.72 + 1.72i)27-s + 1.11i·29-s − 1.87·41-s + (−0.340 − 0.340i)43-s + (0.978 − 0.978i)47-s − 0.428i·49-s + 1.02·61-s + (1.97 + 1.97i)63-s + (1.36 − 1.36i)67-s − 3.61i·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86810 + 1.04153i\)
\(L(\frac12)\) \(\approx\) \(1.86810 + 1.04153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
7 \( 1 + (-2.23 + 2.23i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (6.70 + 6.70i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + (2.23 + 2.23i)T + 43iT^{2} \)
47 \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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

−10.98357474495040166501479501267, −10.39988775557472225409398483677, −9.697404426810671569378454512657, −8.524893452988427596541792083614, −8.163335110556912520481640942854, −6.97309444263489073158465307755, −5.18623256982816384649673695548, −4.32505821517424656020472702238, −3.51443141489327993082822069688, −2.10597507111839493373218313936, 1.61449907431032737715089464787, 2.50440753921127709088745280024, 3.81878661239165248028630308085, 5.50338568631442837052196535536, 6.58946512478013834400918603139, 7.72231431178616882719109864936, 8.180222419607808861878002313188, 9.014301076137729022925882581596, 9.929877628441445663716608336962, 11.60827613628384846112671246941

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