Properties

Degree 2
Conductor 29
Sign $0.191 + 0.981i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.929 − 2.65i)2-s + (−1.45 − 0.164i)3-s + (−3.06 − 2.44i)4-s + (1.35 + 2.82i)5-s + (−1.79 + 3.72i)6-s + (2.26 + 2.83i)7-s + (0.186 − 0.117i)8-s + (−6.67 − 1.52i)9-s + (8.75 − 0.986i)10-s + (9.83 + 6.17i)11-s + (4.07 + 4.07i)12-s + (−18.5 + 4.23i)13-s + (9.64 − 3.37i)14-s + (−1.51 − 4.33i)15-s + (−3.62 − 15.8i)16-s + (−12.3 + 12.3i)17-s + ⋯
L(s)  = 1  + (0.464 − 1.32i)2-s + (−0.486 − 0.0547i)3-s + (−0.766 − 0.611i)4-s + (0.271 + 0.564i)5-s + (−0.298 + 0.620i)6-s + (0.323 + 0.405i)7-s + (0.0232 − 0.0146i)8-s + (−0.741 − 0.169i)9-s + (0.875 − 0.0986i)10-s + (0.893 + 0.561i)11-s + (0.339 + 0.339i)12-s + (−1.42 + 0.325i)13-s + (0.688 − 0.240i)14-s + (−0.101 − 0.289i)15-s + (−0.226 − 0.993i)16-s + (−0.725 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.191 + 0.981i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (21, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.191 + 0.981i)$
$L(\frac{3}{2})$  $\approx$  $0.835184 - 0.687933i$
$L(\frac12)$  $\approx$  $0.835184 - 0.687933i$
$L(2)$   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 \( 1 + (-14.7 + 24.9i)T \)
good2 \( 1 + (-0.929 + 2.65i)T + (-3.12 - 2.49i)T^{2} \)
3 \( 1 + (1.45 + 0.164i)T + (8.77 + 2.00i)T^{2} \)
5 \( 1 + (-1.35 - 2.82i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (-2.26 - 2.83i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-9.83 - 6.17i)T + (52.4 + 109. i)T^{2} \)
13 \( 1 + (18.5 - 4.23i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (12.3 - 12.3i)T - 289iT^{2} \)
19 \( 1 + (2.90 + 25.7i)T + (-351. + 80.3i)T^{2} \)
23 \( 1 + (-9.50 - 4.57i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (8.22 - 23.5i)T + (-751. - 599. i)T^{2} \)
37 \( 1 + (25.3 - 15.9i)T + (593. - 1.23e3i)T^{2} \)
41 \( 1 + (-36.7 - 36.7i)T + 1.68e3iT^{2} \)
43 \( 1 + (-37.8 + 13.2i)T + (1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (-16.7 + 26.6i)T + (-958. - 1.99e3i)T^{2} \)
53 \( 1 + (47.4 - 22.8i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 - 2.51T + 3.48e3T^{2} \)
61 \( 1 + (29.2 + 3.29i)T + (3.62e3 + 828. i)T^{2} \)
67 \( 1 + (127. + 29.0i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-73.2 + 16.7i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-8.81 - 25.2i)T + (-4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-51.2 - 81.5i)T + (-2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (-1.49 + 1.87i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (28.8 - 82.3i)T + (-6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (57.2 - 6.45i)T + (9.17e3 - 2.09e3i)T^{2} \)
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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

−17.06454036002864528350971308486, −15.02668363121099088499877068721, −14.01384751232703878679337279769, −12.46749471705862965640296204399, −11.66235885540234904570687369955, −10.66974875268026286446868491648, −9.238108296749495220750681002899, −6.75616602952994572854723397761, −4.71823541675549611414978035239, −2.50462259204940996241159694855, 4.76604292644190696104409931247, 5.92569786665881547965209700259, 7.43852214682411994380589850704, 8.936757517352593689033203314822, 10.92125803176826115285328061260, 12.43979124264882012430074320426, 14.00052647212486396419338822754, 14.62736550384429850947041300465, 16.21209387655335792388850838816, 16.97463654383482975596298365318

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