Properties

Degree 2
Conductor 29
Sign $0.465 - 0.885i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.488 + 0.776i)2-s + (−1.66 + 4.74i)3-s + (1.37 − 2.84i)4-s + (−1.98 − 0.453i)5-s + (−4.50 + 1.02i)6-s + (9.56 − 4.60i)7-s + (6.52 − 0.735i)8-s + (−12.7 − 10.1i)9-s + (−0.616 − 1.76i)10-s + (−11.6 − 1.31i)11-s + (11.2 + 11.2i)12-s + (−11.0 + 8.79i)13-s + (8.24 + 5.18i)14-s + (5.45 − 8.67i)15-s + (−4.12 − 5.16i)16-s + (0.154 − 0.154i)17-s + ⋯
L(s)  = 1  + (0.244 + 0.388i)2-s + (−0.553 + 1.58i)3-s + (0.342 − 0.711i)4-s + (−0.396 − 0.0906i)5-s + (−0.750 + 0.171i)6-s + (1.36 − 0.657i)7-s + (0.815 − 0.0919i)8-s + (−1.41 − 1.13i)9-s + (−0.0616 − 0.176i)10-s + (−1.05 − 0.119i)11-s + (0.936 + 0.936i)12-s + (−0.848 + 0.676i)13-s + (0.588 + 0.370i)14-s + (0.363 − 0.578i)15-s + (−0.257 − 0.323i)16-s + (0.00910 − 0.00910i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.465 - 0.885i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.465 - 0.885i)$
$L(\frac{3}{2})$  $\approx$  $0.859544 + 0.519135i$
$L(\frac12)$  $\approx$  $0.859544 + 0.519135i$
$L(2)$   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 \( 1 + (15.7 - 24.3i)T \)
good2 \( 1 + (-0.488 - 0.776i)T + (-1.73 + 3.60i)T^{2} \)
3 \( 1 + (1.66 - 4.74i)T + (-7.03 - 5.61i)T^{2} \)
5 \( 1 + (1.98 + 0.453i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (-9.56 + 4.60i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (11.6 + 1.31i)T + (117. + 26.9i)T^{2} \)
13 \( 1 + (11.0 - 8.79i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-0.154 + 0.154i)T - 289iT^{2} \)
19 \( 1 + (-20.3 + 7.12i)T + (282. - 225. i)T^{2} \)
23 \( 1 + (-1.51 - 6.62i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (0.406 + 0.646i)T + (-416. + 865. i)T^{2} \)
37 \( 1 + (6.96 - 0.784i)T + (1.33e3 - 304. i)T^{2} \)
41 \( 1 + (-35.8 - 35.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-18.1 - 11.4i)T + (802. + 1.66e3i)T^{2} \)
47 \( 1 + (2.57 - 22.8i)T + (-2.15e3 - 491. i)T^{2} \)
53 \( 1 + (-12.7 + 55.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + 48.4T + 3.48e3T^{2} \)
61 \( 1 + (24.4 - 69.8i)T + (-2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (-33.7 - 26.9i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (-78.3 + 62.5i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (-5.59 + 8.90i)T + (-2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-4.65 - 41.2i)T + (-6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (-85.5 - 41.1i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (89.2 + 142. i)T + (-3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-20.2 - 57.9i)T + (-7.35e3 + 5.86e3i)T^{2} \)
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

−16.67361498564826132023047509789, −15.86031556613409573045068230384, −14.93922157394777123780629646483, −14.03669515958099033294414169551, −11.50581895660908274251584403118, −10.81102061500491318263156519581, −9.686240590628959892687459855479, −7.58632643514464683914865254181, −5.33475322398810176268929386281, −4.52504164191194408946065898757, 2.26853040547057054155404056923, 5.37205647845159185847971192468, 7.56610913536133841216382978870, 7.914172458501602068251131731395, 11.01946568590702735627916149109, 11.93698043641229834294827352861, 12.59170739203885546360805801045, 13.83770272992953111112053281161, 15.47498200219575080012435292779, 17.16850074450450267801836593695

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