Properties

Degree 2
Conductor 29
Sign $0.999 + 0.0174i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.56 − 0.401i)2-s + (1.78 + 1.12i)3-s + (8.62 + 1.96i)4-s + (5.24 − 4.17i)5-s + (−5.90 − 4.70i)6-s + (2.11 + 9.28i)7-s + (−16.3 − 5.73i)8-s + (−1.97 − 4.11i)9-s + (−20.3 + 12.7i)10-s + (−0.700 + 0.245i)11-s + (13.1 + 13.1i)12-s + (3.39 − 7.04i)13-s + (−3.82 − 33.9i)14-s + (14.0 − 1.58i)15-s + (24.1 + 11.6i)16-s + (−17.4 + 17.4i)17-s + ⋯
L(s)  = 1  + (−1.78 − 0.200i)2-s + (0.594 + 0.373i)3-s + (2.15 + 0.492i)4-s + (1.04 − 0.835i)5-s + (−0.983 − 0.784i)6-s + (0.302 + 1.32i)7-s + (−2.04 − 0.716i)8-s + (−0.219 − 0.456i)9-s + (−2.03 + 1.27i)10-s + (−0.0637 + 0.0222i)11-s + (1.09 + 1.09i)12-s + (0.261 − 0.542i)13-s + (−0.273 − 2.42i)14-s + (0.935 − 0.105i)15-s + (1.51 + 0.727i)16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.999 + 0.0174i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.999 + 0.0174i)$
$L(\frac{3}{2})$  $\approx$  $0.623147 - 0.00544588i$
$L(\frac12)$  $\approx$  $0.623147 - 0.00544588i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \(F_p\) is a polynomial of degree 2. If $p = 29$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 \( 1 + (19.7 + 21.2i)T \)
good2 \( 1 + (3.56 + 0.401i)T + (3.89 + 0.890i)T^{2} \)
3 \( 1 + (-1.78 - 1.12i)T + (3.90 + 8.10i)T^{2} \)
5 \( 1 + (-5.24 + 4.17i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (-2.11 - 9.28i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (0.700 - 0.245i)T + (94.6 - 75.4i)T^{2} \)
13 \( 1 + (-3.39 + 7.04i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (17.4 - 17.4i)T - 289iT^{2} \)
19 \( 1 + (7.96 + 12.6i)T + (-156. + 325. i)T^{2} \)
23 \( 1 + (11.9 - 15.0i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (-0.327 - 0.0369i)T + (936. + 213. i)T^{2} \)
37 \( 1 + (15.2 + 5.34i)T + (1.07e3 + 853. i)T^{2} \)
41 \( 1 + (-28.0 - 28.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (-0.679 - 6.02i)T + (-1.80e3 + 411. i)T^{2} \)
47 \( 1 + (-1.43 - 4.10i)T + (-1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (37.6 + 47.1i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 - 91.1T + 3.48e3T^{2} \)
61 \( 1 + (-6.43 - 4.04i)T + (1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (-29.8 - 62.0i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (38.1 - 79.2i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (29.1 - 3.28i)T + (5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (-7.28 + 20.8i)T + (-4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (-9.19 + 40.2i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (-98.0 - 11.0i)T + (7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (-136. + 86.0i)T + (4.08e3 - 8.47e3i)T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.53534610553738230695584125403, −15.90876703702229667732650243733, −15.01496316256726878822835922114, −12.96155709527904930107817240449, −11.46016721054536833326264770882, −9.876193411912551826500286347404, −8.972834492986334023686666700909, −8.403278286763391354245687720317, −5.99760421440243241711211321872, −2.17403251881833238533724983390, 2.09147568901355725236710768500, 6.63320842727198425524264247209, 7.60532482196887026663776969558, 9.004828155093114908088000586703, 10.33181459063392946598500960859, 11.02591209877167334834463414520, 13.63106037057178340601902540659, 14.43066149536923724815595972320, 16.25641486207767262262849749484, 17.20381420185940539141020432792

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