Properties

Degree 2
Conductor 29
Sign $0.750 - 0.660i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 + 1.29i)2-s + (−2.93 + 1.02i)3-s + (0.827 + 1.71i)4-s + (5.87 − 1.34i)5-s + (−7.36 − 1.67i)6-s + (−9.36 − 4.50i)7-s + (0.569 − 5.05i)8-s + (0.506 − 0.404i)9-s + (13.8 + 4.83i)10-s + (0.977 + 8.67i)11-s + (−4.19 − 4.19i)12-s + (8.98 + 7.16i)13-s + (−13.4 − 21.3i)14-s + (−15.8 + 9.96i)15-s + (12.4 − 15.6i)16-s + (−9.77 + 9.77i)17-s + ⋯
L(s)  = 1  + (1.02 + 0.646i)2-s + (−0.977 + 0.341i)3-s + (0.206 + 0.429i)4-s + (1.17 − 0.268i)5-s + (−1.22 − 0.279i)6-s + (−1.33 − 0.644i)7-s + (0.0711 − 0.631i)8-s + (0.0562 − 0.0448i)9-s + (1.38 + 0.483i)10-s + (0.0888 + 0.788i)11-s + (−0.349 − 0.349i)12-s + (0.691 + 0.551i)13-s + (−0.960 − 1.52i)14-s + (−1.05 + 0.664i)15-s + (0.779 − 0.976i)16-s + (−0.574 + 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.750 - 0.660i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.750 - 0.660i)$
$L(\frac{3}{2})$  $\approx$  $1.14785 + 0.433183i$
$L(\frac12)$  $\approx$  $1.14785 + 0.433183i$
$L(2)$   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 \( 1 + (28.9 + 1.48i)T \)
good2 \( 1 + (-2.05 - 1.29i)T + (1.73 + 3.60i)T^{2} \)
3 \( 1 + (2.93 - 1.02i)T + (7.03 - 5.61i)T^{2} \)
5 \( 1 + (-5.87 + 1.34i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (9.36 + 4.50i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (-0.977 - 8.67i)T + (-117. + 26.9i)T^{2} \)
13 \( 1 + (-8.98 - 7.16i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (9.77 - 9.77i)T - 289iT^{2} \)
19 \( 1 + (4.49 - 12.8i)T + (-282. - 225. i)T^{2} \)
23 \( 1 + (-6.44 + 28.2i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (-32.8 - 20.6i)T + (416. + 865. i)T^{2} \)
37 \( 1 + (-6.61 + 58.7i)T + (-1.33e3 - 304. i)T^{2} \)
41 \( 1 + (-28.8 - 28.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-23.3 - 37.1i)T + (-802. + 1.66e3i)T^{2} \)
47 \( 1 + (16.5 - 1.86i)T + (2.15e3 - 491. i)T^{2} \)
53 \( 1 + (12.8 + 56.5i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 34.0T + 3.48e3T^{2} \)
61 \( 1 + (-5.90 + 2.06i)T + (2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (53.1 - 42.4i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (24.4 + 19.5i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (43.4 - 27.2i)T + (2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-36.9 - 4.16i)T + (6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (-62.9 + 30.2i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-23.7 - 14.9i)T + (3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (33.9 + 11.8i)T + (7.35e3 + 5.86e3i)T^{2} \)
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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

−16.63370031734626087845225747455, −16.10085266833873541250810105564, −14.49314200630331247118845176446, −13.34056958368612604681149829939, −12.60384112745103212991983047438, −10.56836104612440131314766543871, −9.561785284806211974340524121393, −6.58527801647096413483558196111, −5.92273604664680627849655125928, −4.34963896828610810893406508030, 2.98020498858095359387189277077, 5.60425262649109687948534855117, 6.27690633875873339911594560134, 9.211208761656787579719812202007, 10.85578793440681775914991001985, 11.91015046454507410857762500919, 13.17666346820860989176358171566, 13.61312314174545842357691869043, 15.45853731049718275388510214040, 16.96850270222548521347984311408

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