Properties

Degree 2
Conductor 29
Sign $0.913 + 0.407i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.65 − 0.578i)2-s + (−0.186 − 1.65i)3-s + (−0.727 + 0.579i)4-s + (−0.825 + 1.71i)5-s + (−1.26 − 2.62i)6-s + (−1.24 + 1.56i)7-s + (−4.59 + 7.31i)8-s + (6.07 − 1.38i)9-s + (−0.373 + 3.31i)10-s + (−6.63 − 10.5i)11-s + (1.09 + 1.09i)12-s + (−3.80 − 0.868i)13-s + (−1.15 + 3.30i)14-s + (2.98 + 1.04i)15-s + (−2.54 + 11.1i)16-s + (7.59 − 7.59i)17-s + ⋯
L(s)  = 1  + (0.826 − 0.289i)2-s + (−0.0621 − 0.551i)3-s + (−0.181 + 0.144i)4-s + (−0.165 + 0.342i)5-s + (−0.210 − 0.437i)6-s + (−0.178 + 0.223i)7-s + (−0.574 + 0.914i)8-s + (0.674 − 0.154i)9-s + (−0.0373 + 0.331i)10-s + (−0.602 − 0.959i)11-s + (0.0912 + 0.0912i)12-s + (−0.292 − 0.0668i)13-s + (−0.0827 + 0.236i)14-s + (0.199 + 0.0697i)15-s + (−0.158 + 0.695i)16-s + (0.446 − 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.913 + 0.407i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.913 + 0.407i)$
$L(\frac{3}{2})$  $\approx$  $1.20315 - 0.256245i$
$L(\frac12)$  $\approx$  $1.20315 - 0.256245i$
$L(2)$   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 \( 1 + (-8.15 - 27.8i)T \)
good2 \( 1 + (-1.65 + 0.578i)T + (3.12 - 2.49i)T^{2} \)
3 \( 1 + (0.186 + 1.65i)T + (-8.77 + 2.00i)T^{2} \)
5 \( 1 + (0.825 - 1.71i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + (1.24 - 1.56i)T + (-10.9 - 47.7i)T^{2} \)
11 \( 1 + (6.63 + 10.5i)T + (-52.4 + 109. i)T^{2} \)
13 \( 1 + (3.80 + 0.868i)T + (152. + 73.3i)T^{2} \)
17 \( 1 + (-7.59 + 7.59i)T - 289iT^{2} \)
19 \( 1 + (0.137 + 0.0154i)T + (351. + 80.3i)T^{2} \)
23 \( 1 + (-26.7 + 12.8i)T + (329. - 413. i)T^{2} \)
31 \( 1 + (54.1 - 18.9i)T + (751. - 599. i)T^{2} \)
37 \( 1 + (29.9 - 47.6i)T + (-593. - 1.23e3i)T^{2} \)
41 \( 1 + (25.9 + 25.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-5.16 + 14.7i)T + (-1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (-55.9 + 35.1i)T + (958. - 1.99e3i)T^{2} \)
53 \( 1 + (-29.3 - 14.1i)T + (1.75e3 + 2.19e3i)T^{2} \)
59 \( 1 - 0.396T + 3.48e3T^{2} \)
61 \( 1 + (6.99 + 62.1i)T + (-3.62e3 + 828. i)T^{2} \)
67 \( 1 + (21.7 - 4.97i)T + (4.04e3 - 1.94e3i)T^{2} \)
71 \( 1 + (64.8 + 14.7i)T + (4.54e3 + 2.18e3i)T^{2} \)
73 \( 1 + (-76.3 - 26.7i)T + (4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-67.7 - 42.5i)T + (2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (68.4 + 85.8i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (77.7 - 27.2i)T + (6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-19.5 + 173. i)T + (-9.17e3 - 2.09e3i)T^{2} \)
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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

−16.83753293767611771842095039381, −15.35744191249550161358643275556, −14.09393946313299493877267908459, −13.02863400302361874599003254202, −12.20821867456442800486865002371, −10.78736868471982241225272529820, −8.806964114095973891357057442634, −7.14139242802598570942609641989, −5.26814537954540571231719335186, −3.21028728649660128859876654384, 4.10954497117774731565483413246, 5.28535564427513188446412866702, 7.24356589831141414919023073603, 9.370494919291570751956801127243, 10.43968157939932586521769603215, 12.45603964869917797688440304243, 13.25003498915606651007064915554, 14.75369997801927775411263153892, 15.51992463722839861181769819303, 16.62733296614835653656893795338

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