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Properties

Label 2-29e2-29.9-c1-0-27
Degree $2$
Conductor $841$
Sign $0.983 + 0.182i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.179 − 0.373i)2-s + (0.323 − 0.258i)3-s + (1.14 − 1.42i)4-s + (−0.900 + 0.433i)5-s + (−0.154 − 0.0744i)6-s + (1.76 + 2.21i)7-s + (−1.54 − 0.352i)8-s + (−0.629 + 2.75i)9-s + (0.323 + 0.258i)10-s + (2.35 − 0.537i)11-s − 0.757i·12-s + (0.406 + 1.78i)13-s + (0.508 − 1.05i)14-s + (−0.179 + 0.373i)15-s + (−0.667 − 2.92i)16-s − 4.82i·17-s + ⋯
L(s)  = 1  + (−0.127 − 0.263i)2-s + (0.186 − 0.149i)3-s + (0.570 − 0.714i)4-s + (−0.402 + 0.194i)5-s + (−0.0631 − 0.0303i)6-s + (0.666 + 0.835i)7-s + (−0.546 − 0.124i)8-s + (−0.209 + 0.919i)9-s + (0.102 + 0.0816i)10-s + (0.709 − 0.161i)11-s − 0.218i·12-s + (0.112 + 0.494i)13-s + (0.135 − 0.282i)14-s + (−0.0464 + 0.0963i)15-s + (−0.166 − 0.731i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.983 + 0.182i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81804 - 0.167168i\)
\(L(\frac12)\) \(\approx\) \(1.81804 - 0.167168i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.179 + 0.373i)T + (-1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.323 + 0.258i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-1.76 - 2.21i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-2.35 + 0.537i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.406 - 1.78i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.82iT - 17T^{2} \)
19 \( 1 + (-4.69 - 3.74i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-6.89 - 3.32i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-1.76 - 3.66i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (3.89 + 0.890i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (-2.78 + 5.77i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.11 - 1.16i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (-6.74 + 3.24i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (0.647 - 0.516i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (1.25 - 5.51i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (0.705 + 3.09i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.73 - 3.60i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (0.403 + 0.0921i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (2.28 - 2.85i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.94 + 4.04i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (-9.76 - 7.78i)T + (21.5 + 94.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26482350517711504008008562892, −9.250726610255050472195133825952, −8.682333027841706020236168001914, −7.45108402280751584573775270231, −6.95715931583472619112331554019, −5.50279241004033151773693988158, −5.19331497717630156456096825422, −3.50588144736994817605187887393, −2.38963248675196810849217931859, −1.38507396291318721255039666013, 1.09373884185843863104007479617, 2.89494119025293024455054100447, 3.79818754423786043138109197984, 4.63625610296773048736799868495, 6.14277650217566964829832029444, 6.87481825678463983425336278300, 7.74027756056407047135021360936, 8.400705091509846325619577854721, 9.171625290462971495394603053746, 10.23534048148280687539318617294

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