Properties

Degree 2
Conductor 29
Sign $0.541 - 0.840i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0361 + 0.320i)2-s + (−2.40 + 3.82i)3-s + (3.79 + 0.866i)4-s + (1.11 − 0.889i)5-s + (−1.13 − 0.909i)6-s + (−2.06 − 9.04i)7-s + (−0.841 + 2.40i)8-s + (−4.94 − 10.2i)9-s + (0.245 + 0.390i)10-s + (3.02 + 8.63i)11-s + (−12.4 + 12.4i)12-s + (7.25 − 15.0i)13-s + (2.97 − 0.335i)14-s + (0.721 + 6.40i)15-s + (13.2 + 6.40i)16-s + (−13.4 − 13.4i)17-s + ⋯
L(s)  = 1  + (−0.0180 + 0.160i)2-s + (−0.800 + 1.27i)3-s + (0.949 + 0.216i)4-s + (0.223 − 0.177i)5-s + (−0.189 − 0.151i)6-s + (−0.294 − 1.29i)7-s + (−0.105 + 0.300i)8-s + (−0.549 − 1.14i)9-s + (0.0245 + 0.0390i)10-s + (0.274 + 0.785i)11-s + (−1.03 + 1.03i)12-s + (0.558 − 1.15i)13-s + (0.212 − 0.0239i)14-s + (0.0481 + 0.427i)15-s + (0.831 + 0.400i)16-s + (−0.793 − 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.541 - 0.840i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (27, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.541 - 0.840i)$
$L(\frac{3}{2})$  $\approx$  $0.808606 + 0.441201i$
$L(\frac12)$  $\approx$  $0.808606 + 0.441201i$
$L(2)$   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 \( 1 + (-8.81 + 27.6i)T \)
good2 \( 1 + (0.0361 - 0.320i)T + (-3.89 - 0.890i)T^{2} \)
3 \( 1 + (2.40 - 3.82i)T + (-3.90 - 8.10i)T^{2} \)
5 \( 1 + (-1.11 + 0.889i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (2.06 + 9.04i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (-3.02 - 8.63i)T + (-94.6 + 75.4i)T^{2} \)
13 \( 1 + (-7.25 + 15.0i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (13.4 + 13.4i)T + 289iT^{2} \)
19 \( 1 + (6.13 - 3.85i)T + (156. - 325. i)T^{2} \)
23 \( 1 + (26.9 - 33.8i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (-2.27 + 20.1i)T + (-936. - 213. i)T^{2} \)
37 \( 1 + (-5.54 + 15.8i)T + (-1.07e3 - 853. i)T^{2} \)
41 \( 1 + (11.0 - 11.0i)T - 1.68e3iT^{2} \)
43 \( 1 + (1.43 - 0.161i)T + (1.80e3 - 411. i)T^{2} \)
47 \( 1 + (-52.6 + 18.4i)T + (1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-26.2 - 32.9i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 + 40.0T + 3.48e3T^{2} \)
61 \( 1 + (31.5 - 50.2i)T + (-1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-27.0 - 56.0i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (-16.5 + 34.2i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (-10.1 - 90.2i)T + (-5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (68.6 + 24.0i)T + (4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-14.4 + 63.1i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (3.18 - 28.2i)T + (-7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-59.0 - 93.9i)T + (-4.08e3 + 8.47e3i)T^{2} \)
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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

−17.01034474920430227828924595109, −15.89716120705636360247000336514, −15.27906174232742602105953442661, −13.41479671950399744728682640806, −11.70167453848296970205736596642, −10.66033613145820808761404041998, −9.776294329642087303724142691125, −7.44779770868741291703020394846, −5.82114651566775052694556221939, −3.97955499946047638418342504944, 2.09619830703676726421077332286, 6.16412286024120911823177683511, 6.50755498826462301859141793176, 8.605687406308242587672113413205, 10.77435376671243456676199989192, 11.85835395088688539868266295994, 12.55173203444341190926116899808, 14.13337639700709727393029163442, 15.72827284597501244112911423973, 16.74313735842045586589585630827

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