↑

Properties

Label 2-29-29.17-c8-0-15
Degree $2$
Conductor $29$
Sign $0.231 + 0.972i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.384 − 0.384i)2-s + (93.7 − 93.7i)3-s + 255. i·4-s − 802. i·5-s − 72.1i·6-s + 2.72e3·7-s + (197. + 197. i)8-s − 1.10e4i·9-s + (−309. − 309. i)10-s + (−1.03e4 + 1.03e4i)11-s + (2.39e4 + 2.39e4i)12-s − 2.10e4i·13-s + (1.04e3 − 1.04e3i)14-s + (−7.52e4 − 7.52e4i)15-s − 6.53e4·16-s + (7.58e4 − 7.58e4i)17-s + ⋯
L(s)  = 1  + (0.0240 − 0.0240i)2-s + (1.15 − 1.15i)3-s + 0.998i·4-s − 1.28i·5-s − 0.0557i·6-s + 1.13·7-s + (0.0480 + 0.0480i)8-s − 1.67i·9-s + (−0.0309 − 0.0309i)10-s + (−0.705 + 0.705i)11-s + (1.15 + 1.15i)12-s − 0.736i·13-s + (0.0273 − 0.0273i)14-s + (−1.48 − 1.48i)15-s − 0.996·16-s + (0.907 − 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.231 + 0.972i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.15614 - 1.70401i\)
\(L(\frac12)\) \(\approx\) \(2.15614 - 1.70401i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.90e5 + 6.44e5i)T \)
good2 \( 1 + (-0.384 + 0.384i)T - 256iT^{2} \)
3 \( 1 + (-93.7 + 93.7i)T - 6.56e3iT^{2} \)
5 \( 1 + 802. iT - 3.90e5T^{2} \)
7 \( 1 - 2.72e3T + 5.76e6T^{2} \)
11 \( 1 + (1.03e4 - 1.03e4i)T - 2.14e8iT^{2} \)
13 \( 1 + 2.10e4iT - 8.15e8T^{2} \)
17 \( 1 + (-7.58e4 + 7.58e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (-4.94e4 + 4.94e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 1.05e4T + 7.83e10T^{2} \)
31 \( 1 + (1.05e6 - 1.05e6i)T - 8.52e11iT^{2} \)
37 \( 1 + (-2.43e6 - 2.43e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (-3.64e6 - 3.64e6i)T + 7.98e12iT^{2} \)
43 \( 1 + (2.65e6 - 2.65e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-2.80e6 - 2.80e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 5.48e6T + 6.22e13T^{2} \)
59 \( 1 + 4.51e6T + 1.46e14T^{2} \)
61 \( 1 + (9.36e6 - 9.36e6i)T - 1.91e14iT^{2} \)
67 \( 1 - 9.67e6iT - 4.06e14T^{2} \)
71 \( 1 - 5.43e6iT - 6.45e14T^{2} \)
73 \( 1 + (1.88e7 + 1.88e7i)T + 8.06e14iT^{2} \)
79 \( 1 + (-7.39e6 + 7.39e6i)T - 1.51e15iT^{2} \)
83 \( 1 - 6.15e7T + 2.25e15T^{2} \)
89 \( 1 + (-5.11e7 + 5.11e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-6.15e7 - 6.15e7i)T + 7.83e15iT^{2} \)
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

−14.78509866882759736121975837407, −13.42890739088765866721476893180, −12.78178062797388728587549675103, −11.81074008542611948177623012708, −9.190051781874833495665503550529, −7.970023645975126283447869208821, −7.64852170310818324216742265174, −4.82677358412633524521999844947, −2.75896299051710304019985055413, −1.21465858625540056219298633643, 2.17253482613669060075813249123, 3.82109565467686786144502674797, 5.55013288920977427973192807095, 7.69941708212447904241314439761, 9.188918647021391019974394199690, 10.49428028583370057702256748058, 11.02584605770885883853738726940, 13.90608169000643060577354935630, 14.59481302395725198762746253166, 15.00527952931143244613437732642

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