Properties

Degree 2
Conductor 29
Sign $-0.232 - 0.972i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 3.54i)2-s + (4.35 + 0.491i)3-s + (−7.88 − 6.28i)4-s + (−0.825 − 1.71i)5-s + (−7.14 + 14.8i)6-s + (−2.97 − 3.72i)7-s + (19.3 − 12.1i)8-s + (9.98 + 2.27i)9-s + (7.09 − 0.798i)10-s + (8.99 + 5.65i)11-s + (−31.2 − 31.2i)12-s + (−14.5 + 3.31i)13-s + (16.8 − 5.91i)14-s + (−2.75 − 7.87i)15-s + (10.0 + 44.1i)16-s + (−1.89 + 1.89i)17-s + ⋯
L(s)  = 1  + (−0.619 + 1.77i)2-s + (1.45 + 0.163i)3-s + (−1.97 − 1.57i)4-s + (−0.165 − 0.342i)5-s + (−1.19 + 2.47i)6-s + (−0.424 − 0.532i)7-s + (2.41 − 1.51i)8-s + (1.10 + 0.253i)9-s + (0.709 − 0.0798i)10-s + (0.818 + 0.514i)11-s + (−2.60 − 2.60i)12-s + (−1.11 + 0.254i)13-s + (1.20 − 0.422i)14-s + (−0.183 − 0.524i)15-s + (0.629 + 2.75i)16-s + (−0.111 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.232 - 0.972i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (21, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.232 - 0.972i)$
$L(\frac{3}{2})$  $\approx$  $0.577078 + 0.731619i$
$L(\frac12)$  $\approx$  $0.577078 + 0.731619i$
$L(2)$   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 \( 1 + (13.6 - 25.5i)T \)
good2 \( 1 + (1.23 - 3.54i)T + (-3.12 - 2.49i)T^{2} \)
3 \( 1 + (-4.35 - 0.491i)T + (8.77 + 2.00i)T^{2} \)
5 \( 1 + (0.825 + 1.71i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (2.97 + 3.72i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-8.99 - 5.65i)T + (52.4 + 109. i)T^{2} \)
13 \( 1 + (14.5 - 3.31i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (1.89 - 1.89i)T - 289iT^{2} \)
19 \( 1 + (0.860 + 7.63i)T + (-351. + 80.3i)T^{2} \)
23 \( 1 + (12.8 + 6.21i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (5.75 - 16.4i)T + (-751. - 599. i)T^{2} \)
37 \( 1 + (-47.2 + 29.6i)T + (593. - 1.23e3i)T^{2} \)
41 \( 1 + (-52.6 - 52.6i)T + 1.68e3iT^{2} \)
43 \( 1 + (57.3 - 20.0i)T + (1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (14.1 - 22.4i)T + (-958. - 1.99e3i)T^{2} \)
53 \( 1 + (7.05 - 3.39i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 + 16.5T + 3.48e3T^{2} \)
61 \( 1 + (-45.2 - 5.09i)T + (3.62e3 + 828. i)T^{2} \)
67 \( 1 + (9.62 + 2.19i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-70.8 + 16.1i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-26.3 - 75.3i)T + (-4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (34.8 + 55.4i)T + (-2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (-29.5 + 37.0i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (-8.20 + 23.4i)T + (-6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-77.3 + 8.72i)T + (9.17e3 - 2.09e3i)T^{2} \)
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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

−16.90806927072289205295800999878, −16.06121588762550918129684663630, −14.69629025387009204478241135121, −14.41088767400467848486530281554, −13.00173400876417932021269020788, −9.833961150224128950500011450644, −9.083131767895388810220218156326, −7.898062933228875344876026017202, −6.79830576725497219148471434573, −4.44301625842233396693494725245, 2.41121060941770110206489168690, 3.65851457058082759763965118082, 7.83842429889322977546188023201, 9.064586549058561930429233236621, 9.835891936294349842847813730596, 11.48901801057031576220319493123, 12.65294379445279609257366010462, 13.77309478117086423661720328820, 14.93299952382111606676108626368, 17.01917356463446631024494016787

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