Properties

Label 2-29-29.8-c2-0-1
Degree $2$
Conductor $29$
Sign $0.913 - 0.407i$
Analytic cond. $0.790192$
Root an. cond. $0.888927$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(29\)
Sign: $0.913 - 0.407i$
Analytic conductor: \(0.790192\)
Root analytic conductor: \(0.888927\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1),\ 0.913 - 0.407i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20315 + 0.256245i\)
\(L(\frac12)\) \(\approx\) \(1.20315 + 0.256245i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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