Properties

Degree 2
Conductor 29
Sign $0.943 + 0.331i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.144 − 0.412i)2-s + (0.570 − 0.0643i)3-s + (2.97 − 2.37i)4-s + (−1.02 + 2.13i)5-s + (−0.109 − 0.226i)6-s + (−1.74 + 2.18i)7-s + (−2.89 − 1.81i)8-s + (−8.45 + 1.92i)9-s + (1.02 + 0.115i)10-s + (−7.71 + 4.84i)11-s + (1.54 − 1.54i)12-s + (8.82 + 2.01i)13-s + (1.15 + 0.404i)14-s + (−0.449 + 1.28i)15-s + (3.05 − 13.3i)16-s + (5.09 + 5.09i)17-s + ⋯
L(s)  = 1  + (−0.0722 − 0.206i)2-s + (0.190 − 0.0214i)3-s + (0.744 − 0.593i)4-s + (−0.205 + 0.426i)5-s + (−0.0181 − 0.0377i)6-s + (−0.249 + 0.312i)7-s + (−0.361 − 0.227i)8-s + (−0.939 + 0.214i)9-s + (0.102 + 0.0115i)10-s + (−0.701 + 0.440i)11-s + (0.128 − 0.128i)12-s + (0.679 + 0.155i)13-s + (0.0825 + 0.0288i)14-s + (−0.0299 + 0.0856i)15-s + (0.191 − 0.837i)16-s + (0.299 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.943 + 0.331i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (18, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.943 + 0.331i)$
$L(\frac{3}{2})$  $\approx$  $1.00744 - 0.171796i$
$L(\frac12)$  $\approx$  $1.00744 - 0.171796i$
$L(2)$   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 \( 1 + (26.8 - 10.9i)T \)
good2 \( 1 + (0.144 + 0.412i)T + (-3.12 + 2.49i)T^{2} \)
3 \( 1 + (-0.570 + 0.0643i)T + (8.77 - 2.00i)T^{2} \)
5 \( 1 + (1.02 - 2.13i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + (1.74 - 2.18i)T + (-10.9 - 47.7i)T^{2} \)
11 \( 1 + (7.71 - 4.84i)T + (52.4 - 109. i)T^{2} \)
13 \( 1 + (-8.82 - 2.01i)T + (152. + 73.3i)T^{2} \)
17 \( 1 + (-5.09 - 5.09i)T + 289iT^{2} \)
19 \( 1 + (-2.57 + 22.8i)T + (-351. - 80.3i)T^{2} \)
23 \( 1 + (-17.8 + 8.59i)T + (329. - 413. i)T^{2} \)
31 \( 1 + (10.2 + 29.3i)T + (-751. + 599. i)T^{2} \)
37 \( 1 + (-5.78 - 3.63i)T + (593. + 1.23e3i)T^{2} \)
41 \( 1 + (-9.53 + 9.53i)T - 1.68e3iT^{2} \)
43 \( 1 + (-47.8 - 16.7i)T + (1.44e3 + 1.15e3i)T^{2} \)
47 \( 1 + (44.9 + 71.5i)T + (-958. + 1.99e3i)T^{2} \)
53 \( 1 + (-76.8 - 36.9i)T + (1.75e3 + 2.19e3i)T^{2} \)
59 \( 1 - 54.7T + 3.48e3T^{2} \)
61 \( 1 + (85.8 - 9.67i)T + (3.62e3 - 828. i)T^{2} \)
67 \( 1 + (70.4 - 16.0i)T + (4.04e3 - 1.94e3i)T^{2} \)
71 \( 1 + (85.2 + 19.4i)T + (4.54e3 + 2.18e3i)T^{2} \)
73 \( 1 + (41.8 - 119. i)T + (-4.16e3 - 3.32e3i)T^{2} \)
79 \( 1 + (29.8 - 47.4i)T + (-2.70e3 - 5.62e3i)T^{2} \)
83 \( 1 + (-57.5 - 72.2i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (31.8 + 91.0i)T + (-6.19e3 + 4.93e3i)T^{2} \)
97 \( 1 + (-17.8 - 2.00i)T + (9.17e3 + 2.09e3i)T^{2} \)
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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

−16.71544378716972336721202297490, −15.43216308101990612821135128450, −14.69195373399224200737649423759, −13.13125177079054067867320135817, −11.49869516019213228945662804944, −10.67547337472292182427213663832, −9.061353514559118160416223920121, −7.23766000601058518860849177993, −5.66691463526300846629847198970, −2.76279595876678456465117455869, 3.28148896817635677017620667670, 5.89051430759619168346592867433, 7.66701202108481273417462217425, 8.769897329414042059408519299958, 10.75405192163814550960922638866, 11.96287917991187897658284988319, 13.20609769521001490290576134333, 14.67900784141842300040251670416, 16.07305901113780327764535134585, 16.65983489500085614683545864788

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