Properties

Degree 2
Conductor 29
Sign $-0.889 - 0.457i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.887 + 2.53i)2-s + (−5.44 − 0.613i)3-s + (−2.51 − 2.00i)4-s + (3.04 + 6.31i)5-s + (6.38 − 13.2i)6-s + (−0.484 − 0.608i)7-s + (−1.77 + 1.11i)8-s + (20.5 + 4.68i)9-s + (−18.7 + 2.10i)10-s + (4.24 + 2.66i)11-s + (12.4 + 12.4i)12-s + (−2.96 + 0.675i)13-s + (1.97 − 0.690i)14-s + (−12.6 − 36.2i)15-s + (−4.11 − 18.0i)16-s + (5.44 − 5.44i)17-s + ⋯
L(s)  = 1  + (−0.443 + 1.26i)2-s + (−1.81 − 0.204i)3-s + (−0.629 − 0.502i)4-s + (0.608 + 1.26i)5-s + (1.06 − 2.21i)6-s + (−0.0692 − 0.0868i)7-s + (−0.221 + 0.139i)8-s + (2.27 + 0.520i)9-s + (−1.87 + 0.210i)10-s + (0.386 + 0.242i)11-s + (1.04 + 1.04i)12-s + (−0.227 + 0.0519i)13-s + (0.140 − 0.0493i)14-s + (−0.846 − 2.41i)15-s + (−0.257 − 1.12i)16-s + (0.320 − 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.889 - 0.457i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (21, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.889 - 0.457i)$
$L(\frac{3}{2})$  $\approx$  $0.115908 + 0.478291i$
$L(\frac12)$  $\approx$  $0.115908 + 0.478291i$
$L(2)$   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 \( 1 + (-10.4 - 27.0i)T \)
good2 \( 1 + (0.887 - 2.53i)T + (-3.12 - 2.49i)T^{2} \)
3 \( 1 + (5.44 + 0.613i)T + (8.77 + 2.00i)T^{2} \)
5 \( 1 + (-3.04 - 6.31i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (0.484 + 0.608i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-4.24 - 2.66i)T + (52.4 + 109. i)T^{2} \)
13 \( 1 + (2.96 - 0.675i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (-5.44 + 5.44i)T - 289iT^{2} \)
19 \( 1 + (-1.87 - 16.6i)T + (-351. + 80.3i)T^{2} \)
23 \( 1 + (-16.4 - 7.94i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (-7.35 + 21.0i)T + (-751. - 599. i)T^{2} \)
37 \( 1 + (-6.78 + 4.26i)T + (593. - 1.23e3i)T^{2} \)
41 \( 1 + (35.8 + 35.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-61.1 + 21.3i)T + (1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (23.5 - 37.5i)T + (-958. - 1.99e3i)T^{2} \)
53 \( 1 + (-21.6 + 10.4i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 + 90.6T + 3.48e3T^{2} \)
61 \( 1 + (-10.2 - 1.15i)T + (3.62e3 + 828. i)T^{2} \)
67 \( 1 + (-20.7 - 4.72i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-88.7 + 20.2i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-2.59 - 7.42i)T + (-4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-13.3 - 21.2i)T + (-2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (81.3 - 101. i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (34.1 - 97.6i)T + (-6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-113. + 12.7i)T + (9.17e3 - 2.09e3i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.28375811612547060116266482283, −16.57906683026198775816228432087, −15.36460085628772100907750220722, −14.12648729804952338599039756244, −12.24230889990939474381673207677, −10.99695273805420118514334083498, −9.813024819286233933712520848161, −7.29492256370637240411924601364, −6.52324158515107119291765474289, −5.48211178461277939062370115223, 0.983428657714887671064535433412, 4.76474469807522527102497959662, 6.20884573687930653500253430239, 9.139039350866492042586549247649, 10.20116546527916936522137296263, 11.34959684930106334998422859010, 12.26859352666814271228168291008, 13.06630130188958455734554003375, 15.71073688929849530270657527460, 16.93754062207493273289919755293

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