Properties

Degree 2
Conductor 29
Sign $0.162 - 0.986i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.20 − 0.770i)2-s + (−0.552 + 4.90i)3-s + (1.12 + 0.897i)4-s + (2.64 + 5.48i)5-s + (4.99 − 10.3i)6-s + (−3.22 − 4.04i)7-s + (3.17 + 5.05i)8-s + (−14.9 − 3.41i)9-s + (−1.59 − 14.1i)10-s + (3.16 − 5.03i)11-s + (−5.02 + 5.02i)12-s + (14.6 − 3.35i)13-s + (3.98 + 11.4i)14-s + (−28.3 + 9.92i)15-s + (−4.38 − 19.1i)16-s + (22.0 + 22.0i)17-s + ⋯
L(s)  = 1  + (−1.10 − 0.385i)2-s + (−0.184 + 1.63i)3-s + (0.281 + 0.224i)4-s + (0.528 + 1.09i)5-s + (0.832 − 1.72i)6-s + (−0.461 − 0.578i)7-s + (0.397 + 0.632i)8-s + (−1.66 − 0.379i)9-s + (−0.159 − 1.41i)10-s + (0.287 − 0.457i)11-s + (−0.418 + 0.418i)12-s + (1.12 − 0.257i)13-s + (0.284 + 0.814i)14-s + (−1.89 + 0.661i)15-s + (−0.273 − 1.19i)16-s + (1.29 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.162 - 0.986i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.162 - 0.986i)$
$L(\frac{3}{2})$  $\approx$  $0.426548 + 0.361894i$
$L(\frac12)$  $\approx$  $0.426548 + 0.361894i$
$L(2)$   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 \( 1 + (18.7 + 22.1i)T \)
good2 \( 1 + (2.20 + 0.770i)T + (3.12 + 2.49i)T^{2} \)
3 \( 1 + (0.552 - 4.90i)T + (-8.77 - 2.00i)T^{2} \)
5 \( 1 + (-2.64 - 5.48i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (3.22 + 4.04i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-3.16 + 5.03i)T + (-52.4 - 109. i)T^{2} \)
13 \( 1 + (-14.6 + 3.35i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (-22.0 - 22.0i)T + 289iT^{2} \)
19 \( 1 + (0.835 - 0.0941i)T + (351. - 80.3i)T^{2} \)
23 \( 1 + (21.1 + 10.1i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (-21.3 - 7.47i)T + (751. + 599. i)T^{2} \)
37 \( 1 + (-4.25 - 6.77i)T + (-593. + 1.23e3i)T^{2} \)
41 \( 1 + (8.24 - 8.24i)T - 1.68e3iT^{2} \)
43 \( 1 + (3.09 + 8.85i)T + (-1.44e3 + 1.15e3i)T^{2} \)
47 \( 1 + (22.1 + 13.8i)T + (958. + 1.99e3i)T^{2} \)
53 \( 1 + (-47.7 + 22.9i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 - 17.4T + 3.48e3T^{2} \)
61 \( 1 + (-6.87 + 60.9i)T + (-3.62e3 - 828. i)T^{2} \)
67 \( 1 + (26.2 + 5.98i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (4.93 - 1.12i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (53.0 - 18.5i)T + (4.16e3 - 3.32e3i)T^{2} \)
79 \( 1 + (-74.2 + 46.6i)T + (2.70e3 - 5.62e3i)T^{2} \)
83 \( 1 + (48.3 - 60.6i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (138. + 48.3i)T + (6.19e3 + 4.93e3i)T^{2} \)
97 \( 1 + (-0.115 - 1.02i)T + (-9.17e3 + 2.09e3i)T^{2} \)
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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

−17.10513273517139063054860318358, −16.33946563942152055590243135396, −14.90726409178240277018202603002, −13.87942139924366628222816395851, −11.25006303353420560776595435654, −10.27904832812779972790662676191, −9.990707666641201934056455202139, −8.420951407561009741236251805483, −5.98386642003841276409960783258, −3.62614967844496432176362640108, 1.25813071232847850378125019971, 5.89759595459962774344640853819, 7.32050115878989972731845398575, 8.561423302314920130842563474230, 9.574976024816733375830569462436, 11.92326703250144413975478445754, 12.87817157476799522471884562830, 13.77672860832180341520937947416, 16.09345431609769369877648589463, 16.91999978757661817862280584582

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