Properties

Degree 2
Conductor 29
Sign $0.638 + 0.769i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 − 0.811i)2-s + (2.15 − 0.752i)3-s + (−0.725 − 1.50i)4-s + (3.36 − 0.767i)5-s + (−3.38 − 0.773i)6-s + (−0.255 − 0.123i)7-s + (−0.968 + 8.59i)8-s + (−2.97 + 2.37i)9-s + (−4.96 − 1.73i)10-s + (1.62 + 14.4i)11-s + (−2.69 − 2.69i)12-s + (−4.30 − 3.43i)13-s + (0.230 + 0.366i)14-s + (6.65 − 4.18i)15-s + (4.06 − 5.09i)16-s + (5.25 − 5.25i)17-s + ⋯
L(s)  = 1  + (−0.645 − 0.405i)2-s + (0.716 − 0.250i)3-s + (−0.181 − 0.376i)4-s + (0.672 − 0.153i)5-s + (−0.564 − 0.128i)6-s + (−0.0365 − 0.0175i)7-s + (−0.121 + 1.07i)8-s + (−0.330 + 0.263i)9-s + (−0.496 − 0.173i)10-s + (0.147 + 1.31i)11-s + (−0.224 − 0.224i)12-s + (−0.331 − 0.264i)13-s + (0.0164 + 0.0261i)14-s + (0.443 − 0.278i)15-s + (0.253 − 0.318i)16-s + (0.309 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.638 + 0.769i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.638 + 0.769i)$
$L(\frac{3}{2})$  $\approx$  $0.785214 - 0.368712i$
$L(\frac12)$  $\approx$  $0.785214 - 0.368712i$
$L(2)$   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 \( 1 + (-2.32 + 28.9i)T \)
good2 \( 1 + (1.29 + 0.811i)T + (1.73 + 3.60i)T^{2} \)
3 \( 1 + (-2.15 + 0.752i)T + (7.03 - 5.61i)T^{2} \)
5 \( 1 + (-3.36 + 0.767i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (0.255 + 0.123i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (-1.62 - 14.4i)T + (-117. + 26.9i)T^{2} \)
13 \( 1 + (4.30 + 3.43i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (-5.25 + 5.25i)T - 289iT^{2} \)
19 \( 1 + (-9.56 + 27.3i)T + (-282. - 225. i)T^{2} \)
23 \( 1 + (6.05 - 26.5i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (12.9 + 8.13i)T + (416. + 865. i)T^{2} \)
37 \( 1 + (-3.05 + 27.1i)T + (-1.33e3 - 304. i)T^{2} \)
41 \( 1 + (45.8 + 45.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-35.9 - 57.2i)T + (-802. + 1.66e3i)T^{2} \)
47 \( 1 + (-17.2 + 1.94i)T + (2.15e3 - 491. i)T^{2} \)
53 \( 1 + (-1.31 - 5.74i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 43.1T + 3.48e3T^{2} \)
61 \( 1 + (-104. + 36.6i)T + (2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (67.4 - 53.7i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (-45.1 - 36.0i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (42.0 - 26.4i)T + (2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-96.9 - 10.9i)T + (6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (0.956 - 0.460i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-73.6 - 46.2i)T + (3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-54.6 - 19.1i)T + (7.35e3 + 5.86e3i)T^{2} \)
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

−17.32069491645425100754151473007, −15.32328571713014875358297870437, −14.19594789449660392059624167243, −13.24997571189410904408118477451, −11.48973416065095789675545724457, −9.881759917484207708094080712341, −9.175641038557165114019145967336, −7.57894634227678608785125029456, −5.30783691137417245000378221649, −2.17461356851085325180502795350, 3.43132571465032652387404096427, 6.19220692096271339542401984867, 8.076113255426539016348120229707, 9.010328069823498036991138808595, 10.18256031598930259765344455202, 12.16054559921068263722670289020, 13.71765019770498667703813087266, 14.56413430881235410062541524901, 16.19154670989613146986927138345, 16.97982848506661184394450180649

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