Properties

Degree 2
Conductor 29
Sign $0.952 - 0.303i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 + 0.907i)2-s + (1.22 − 0.430i)3-s + (−0.472 − 0.980i)4-s + (−8.15 + 1.86i)5-s + (2.16 + 0.494i)6-s + (7.53 + 3.62i)7-s + (0.972 − 8.62i)8-s + (−5.71 + 4.55i)9-s + (−13.4 − 4.71i)10-s + (−1.08 − 9.61i)11-s + (−1.00 − 1.00i)12-s + (8.88 + 7.08i)13-s + (7.59 + 12.0i)14-s + (−9.22 + 5.79i)15-s + (6.52 − 8.17i)16-s + (11.2 − 11.2i)17-s + ⋯
L(s)  = 1  + (0.722 + 0.453i)2-s + (0.409 − 0.143i)3-s + (−0.118 − 0.245i)4-s + (−1.63 + 0.372i)5-s + (0.361 + 0.0824i)6-s + (1.07 + 0.518i)7-s + (0.121 − 1.07i)8-s + (−0.634 + 0.505i)9-s + (−1.34 − 0.471i)10-s + (−0.0985 − 0.874i)11-s + (−0.0835 − 0.0835i)12-s + (0.683 + 0.544i)13-s + (0.542 + 0.863i)14-s + (−0.615 + 0.386i)15-s + (0.407 − 0.511i)16-s + (0.664 − 0.664i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.952 - 0.303i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.952 - 0.303i)$
$L(\frac{3}{2})$  $\approx$  $1.22243 + 0.189698i$
$L(\frac12)$  $\approx$  $1.22243 + 0.189698i$
$L(2)$   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 \( 1 + (25.9 + 13.0i)T \)
good2 \( 1 + (-1.44 - 0.907i)T + (1.73 + 3.60i)T^{2} \)
3 \( 1 + (-1.22 + 0.430i)T + (7.03 - 5.61i)T^{2} \)
5 \( 1 + (8.15 - 1.86i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (-7.53 - 3.62i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (1.08 + 9.61i)T + (-117. + 26.9i)T^{2} \)
13 \( 1 + (-8.88 - 7.08i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (-11.2 + 11.2i)T - 289iT^{2} \)
19 \( 1 + (5.31 - 15.1i)T + (-282. - 225. i)T^{2} \)
23 \( 1 + (2.98 - 13.0i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (-1.32 - 0.832i)T + (416. + 865. i)T^{2} \)
37 \( 1 + (-2.36 + 21.0i)T + (-1.33e3 - 304. i)T^{2} \)
41 \( 1 + (0.193 + 0.193i)T + 1.68e3iT^{2} \)
43 \( 1 + (38.8 + 61.7i)T + (-802. + 1.66e3i)T^{2} \)
47 \( 1 + (-42.6 + 4.80i)T + (2.15e3 - 491. i)T^{2} \)
53 \( 1 + (-17.1 - 75.1i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 23.4T + 3.48e3T^{2} \)
61 \( 1 + (-56.8 + 19.9i)T + (2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (48.7 - 38.9i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (-79.7 - 63.6i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (23.3 - 14.6i)T + (2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-11.7 - 1.32i)T + (6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (-0.116 + 0.0561i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-24.8 - 15.6i)T + (3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-23.5 - 8.24i)T + (7.35e3 + 5.86e3i)T^{2} \)
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

−16.47700970734770024284798083519, −15.44970891600024275400291579141, −14.54764304054431060548846948110, −13.75327401319994929055944391586, −11.91202843386649214867255805270, −11.00006840480827223645099901176, −8.608632345424827545875160992089, −7.55482264623495055012223862780, −5.52117877308028051883101669967, −3.78722924905628354266105383692, 3.58183116878790893713541986054, 4.70592833656599000456492923497, 7.78728809558735808702452147967, 8.523136325544886572080920247478, 11.02144328299847367878278722354, 11.89333752173594157687574444513, 12.96610993263018244321276221853, 14.52927645879708570226350786435, 15.20636925562374673640843705007, 16.83003002608589983728283921146

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