↑

Properties

Label 2-490-5.4-c1-0-8
Degree $2$
Conductor $490$
Sign $0.994 + 0.100i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
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  − i·2-s + 2.44i·3-s − 4-s + (0.224 − 2.22i)5-s + 2.44·6-s + i·8-s − 2.99·9-s + (−2.22 − 0.224i)10-s + 4.89·11-s − 2.44i·12-s − 0.449i·13-s + (5.44 + 0.550i)15-s + 16-s − 2i·17-s + 2.99i·18-s + 6.44·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (0.100 − 0.994i)5-s + 0.999·6-s + 0.353i·8-s − 0.999·9-s + (−0.703 − 0.0710i)10-s + 1.47·11-s − 0.707i·12-s − 0.124i·13-s + (1.40 + 0.142i)15-s + 0.250·16-s − 0.485i·17-s + 0.707i·18-s + 1.47·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49110 - 0.0751248i\)
\(L(\frac12)\) \(\approx\) \(1.49110 - 0.0751248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.224 + 2.22i)T \)
7 \( 1 \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 0.449iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 - 6.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 8.89iT - 43T^{2} \)
47 \( 1 - 0.898iT - 47T^{2} \)
53 \( 1 - 1.10iT - 53T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 + 8.44T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.89iT - 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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

−10.94606605111687455070055175689, −9.799247521051078332184901168992, −9.398884655668764404338589796385, −8.859846258882581693105894371048, −7.53297166757971703566699406231, −5.82524168539910398593475117643, −4.94241521212648658785106116413, −4.12873622388879084565059835855, −3.26684341551782606088953771249, −1.28296491791730939813381909729, 1.26772411955896465249025104503, 2.81336266714014496872473979834, 4.23042896156416899809120109843, 5.95584051141790940636650169868, 6.48151124622389409407898717838, 7.19236681637688387315842701221, 7.906338874764839013728995370057, 8.979923032469974288378310796445, 9.953602005449436349280177583671, 11.17116653368666286274137107340

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