↑

Properties

Label 2-29e2-29.20-c1-0-10
Degree $2$
Conductor $841$
Sign $0.973 - 0.230i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
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  + (−0.556 + 0.268i)2-s + (−0.385 − 0.483i)3-s + (−1.00 + 1.26i)4-s + (−3.47 + 1.67i)5-s + (0.344 + 0.165i)6-s + (−1.39 − 1.74i)7-s + (0.497 − 2.18i)8-s + (0.582 − 2.55i)9-s + (1.48 − 1.86i)10-s + (0.307 + 1.34i)11-s + 12-s + (0.0525 + 0.230i)13-s + (1.24 + 0.599i)14-s + (2.14 + 1.03i)15-s + (−0.412 − 1.80i)16-s − 4.38·17-s + ⋯
L(s)  = 1  + (−0.393 + 0.189i)2-s + (−0.222 − 0.278i)3-s + (−0.504 + 0.632i)4-s + (−1.55 + 0.747i)5-s + (0.140 + 0.0676i)6-s + (−0.526 − 0.660i)7-s + (0.175 − 0.770i)8-s + (0.194 − 0.850i)9-s + (0.469 − 0.588i)10-s + (0.0927 + 0.406i)11-s + 0.288·12-s + (0.0145 + 0.0638i)13-s + (0.332 + 0.160i)14-s + (0.554 + 0.266i)15-s + (−0.103 − 0.451i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492690 + 0.0575577i\)
\(L(\frac12)\) \(\approx\) \(0.492690 + 0.0575577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (0.556 - 0.268i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (0.385 + 0.483i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (3.47 - 1.67i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.39 + 1.74i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.307 - 1.34i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.0525 - 0.230i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.38T + 17T^{2} \)
19 \( 1 + (3.02 - 3.79i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-1.11 - 0.536i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-9.09 + 4.37i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (1.04 - 4.59i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 3.85T + 41T^{2} \)
43 \( 1 + (-6.51 - 3.13i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.55 - 6.82i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.80 + 0.867i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (0.385 + 0.483i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.339 + 1.48i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (2.33 + 10.2i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-12.3 - 5.94i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-1.35 + 5.93i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (6.20 - 7.77i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.24 - 2.04i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-2.22 + 2.78i)T + (-21.5 - 94.5i)T^{2} \)
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

−10.17333468567448912095392189655, −9.341239925464861039400647367827, −8.308593250661852781774420457237, −7.68056195013240521035390817190, −6.87732693963300807417524174889, −6.43572562836321539502692952725, −4.28712737180342224394427025342, −4.03784564238360633543929468019, −2.98507744484102649545987952670, −0.56226121772665524297482685135, 0.63118108153410745310510595568, 2.46486924639989070843017133410, 4.08700952240942390392367118532, 4.69684914474254772634402347461, 5.53491290433759087902666325059, 6.76561445005449166260024544599, 7.936161182267056858643723640835, 8.735486677518784825368171223503, 9.028874431407587236598256803753, 10.26083555455323772640853894130

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