Properties

Degree 2
Conductor 29
Sign $0.761 + 0.647i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 − 0.713i)2-s + (0.187 − 0.534i)3-s + (1.42 − 2.96i)4-s + (4.21 + 0.962i)5-s + (−0.465 + 0.106i)6-s + (−10.1 + 4.88i)7-s + (−6.10 + 0.688i)8-s + (6.78 + 5.41i)9-s + (−1.20 − 3.44i)10-s + (1.54 + 0.173i)11-s + (−1.31 − 1.31i)12-s + (−11.3 + 9.02i)13-s + (8.03 + 5.04i)14-s + (1.30 − 2.07i)15-s + (−4.97 − 6.23i)16-s + (16.8 − 16.8i)17-s + ⋯
L(s)  = 1  + (−0.224 − 0.356i)2-s + (0.0623 − 0.178i)3-s + (0.356 − 0.740i)4-s + (0.843 + 0.192i)5-s + (−0.0776 + 0.0177i)6-s + (−1.44 + 0.697i)7-s + (−0.763 + 0.0860i)8-s + (0.753 + 0.601i)9-s + (−0.120 − 0.344i)10-s + (0.140 + 0.0157i)11-s + (−0.109 − 0.109i)12-s + (−0.870 + 0.694i)13-s + (0.574 + 0.360i)14-s + (0.0869 − 0.138i)15-s + (−0.310 − 0.389i)16-s + (0.990 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.761 + 0.647i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.761 + 0.647i)$
$L(\frac{3}{2})$  $\approx$  $0.895880 - 0.329478i$
$L(\frac12)$  $\approx$  $0.895880 - 0.329478i$
$L(2)$   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 \( 1 + (20.2 + 20.7i)T \)
good2 \( 1 + (0.448 + 0.713i)T + (-1.73 + 3.60i)T^{2} \)
3 \( 1 + (-0.187 + 0.534i)T + (-7.03 - 5.61i)T^{2} \)
5 \( 1 + (-4.21 - 0.962i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (10.1 - 4.88i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-1.54 - 0.173i)T + (117. + 26.9i)T^{2} \)
13 \( 1 + (11.3 - 9.02i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-16.8 + 16.8i)T - 289iT^{2} \)
19 \( 1 + (-0.448 + 0.157i)T + (282. - 225. i)T^{2} \)
23 \( 1 + (-1.55 - 6.79i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (7.88 + 12.5i)T + (-416. + 865. i)T^{2} \)
37 \( 1 + (-26.2 + 2.96i)T + (1.33e3 - 304. i)T^{2} \)
41 \( 1 + (-46.1 - 46.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (53.7 + 33.7i)T + (802. + 1.66e3i)T^{2} \)
47 \( 1 + (-5.72 + 50.7i)T + (-2.15e3 - 491. i)T^{2} \)
53 \( 1 + (9.09 - 39.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 23.8T + 3.48e3T^{2} \)
61 \( 1 + (-23.8 + 68.2i)T + (-2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (-89.8 - 71.6i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (54.7 - 43.6i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (43.5 - 69.3i)T + (-2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-12.9 - 114. i)T + (-6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (1.11 + 0.537i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (35.1 + 55.9i)T + (-3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (26.0 + 74.3i)T + (-7.35e3 + 5.86e3i)T^{2} \)
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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

−16.67984733203760040029202454686, −15.63395952112570903291704943856, −14.28150586792387692514690363226, −13.02996580747536615441753619294, −11.71486639997820508220821312607, −9.802342718810230226283374858022, −9.677062492549239675363589616453, −6.95937169255699840165804118049, −5.66115803556563025300955777039, −2.39585652968535222842806348355, 3.47710464042977309130130828067, 6.19925671364488136911232974425, 7.43568178139350007933815925245, 9.349995676823830791360931829897, 10.24243525533813367622202227383, 12.49865590055339195104800168366, 13.02917717468862243703181768199, 14.82821396054989206034015834524, 16.14828737154175583209151110789, 16.90368567272095579466893510064

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