Properties

Degree 2
Conductor 29
Sign $0.999 - 0.00995i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.58 − 0.290i)2-s + (−2.11 + 1.32i)3-s + (2.68 − 0.611i)4-s + (−2.49 − 1.98i)5-s + (−5.07 + 4.04i)6-s + (1.30 − 5.70i)7-s + (−3.06 + 1.07i)8-s + (−1.20 + 2.49i)9-s + (−7.00 − 4.40i)10-s + (16.1 + 5.63i)11-s + (−4.85 + 4.85i)12-s + (2.84 + 5.90i)13-s + (1.70 − 15.1i)14-s + (7.90 + 0.890i)15-s + (−17.5 + 8.43i)16-s + (−14.7 − 14.7i)17-s + ⋯
L(s)  = 1  + (1.29 − 0.145i)2-s + (−0.704 + 0.442i)3-s + (0.670 − 0.152i)4-s + (−0.498 − 0.397i)5-s + (−0.845 + 0.674i)6-s + (0.185 − 0.814i)7-s + (−0.383 + 0.134i)8-s + (−0.133 + 0.276i)9-s + (−0.700 − 0.440i)10-s + (1.46 + 0.512i)11-s + (−0.404 + 0.404i)12-s + (0.218 + 0.453i)13-s + (0.121 − 1.07i)14-s + (0.526 + 0.0593i)15-s + (−1.09 + 0.527i)16-s + (−0.869 − 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00995i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00995i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.999 - 0.00995i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.999 - 0.00995i)$
$L(\frac{3}{2})$  $\approx$  $1.32595 + 0.00660053i$
$L(\frac12)$  $\approx$  $1.32595 + 0.00660053i$
$L(2)$   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 \( 1 + (19.9 - 21.0i)T \)
good2 \( 1 + (-2.58 + 0.290i)T + (3.89 - 0.890i)T^{2} \)
3 \( 1 + (2.11 - 1.32i)T + (3.90 - 8.10i)T^{2} \)
5 \( 1 + (2.49 + 1.98i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (-1.30 + 5.70i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (-16.1 - 5.63i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (-2.84 - 5.90i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (14.7 + 14.7i)T + 289iT^{2} \)
19 \( 1 + (-9.65 + 15.3i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (-18.5 - 23.2i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (13.3 - 1.50i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (-17.2 + 6.03i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (44.4 - 44.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (-7.00 + 62.1i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (16.6 - 47.6i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-12.0 + 15.1i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 - 47.9T + 3.48e3T^{2} \)
61 \( 1 + (-41.0 + 25.8i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (6.15 - 12.7i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (11.6 + 24.1i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (12.2 + 1.38i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-33.5 - 95.7i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (9.73 + 42.6i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-53.9 + 6.07i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (124. + 78.0i)T + (4.08e3 + 8.47e3i)T^{2} \)
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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

−16.71415258439123521491834405820, −15.59460288462263593963777272110, −14.27081300139204738045839386687, −13.30929978519129755581951886563, −11.78125827556992088115047951926, −11.22690950257348179035476213291, −9.184572673597639837978945182245, −6.88423805037368216965964737566, −5.02743488751651768053618951073, −4.05174166829846369606229160133, 3.70238518447027919563140712172, 5.69523961997338967315373210074, 6.64677035675330715675302602732, 8.881556932938386183527931466133, 11.32538988280355996251647432730, 11.99943701937965487044543104198, 13.08394780594762164857936378501, 14.62701279721398796943817068148, 15.19030158460250529639510460821, 16.79005631292653498591011943990

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