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  :  $\chi_{29} (14, \cdot )$
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

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

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