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Properties

Label 2-29-29.28-c5-0-1
Degree $2$
Conductor $29$
Sign $-0.684 - 0.729i$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0831i·2-s + 25.1i·3-s + 31.9·4-s − 90.4·5-s − 2.08·6-s − 6.31·7-s + 5.31i·8-s − 388.·9-s − 7.51i·10-s + 505. i·11-s + 804. i·12-s + 275.·13-s − 0.524i·14-s − 2.27e3i·15-s + 1.02e3·16-s + 1.48e3i·17-s + ⋯
L(s)  = 1  + 0.0146i·2-s + 1.61i·3-s + 0.999·4-s − 1.61·5-s − 0.0236·6-s − 0.0486·7-s + 0.0293i·8-s − 1.60·9-s − 0.0237i·10-s + 1.25i·11-s + 1.61i·12-s + 0.451·13-s − 0.000715i·14-s − 2.60i·15-s + 0.999·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.684 - 0.729i$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ -0.684 - 0.729i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.510629 + 1.17905i\)
\(L(\frac12)\) \(\approx\) \(0.510629 + 1.17905i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.09e3 - 3.30e3i)T \)
good2 \( 1 - 0.0831iT - 32T^{2} \)
3 \( 1 - 25.1iT - 243T^{2} \)
5 \( 1 + 90.4T + 3.12e3T^{2} \)
7 \( 1 + 6.31T + 1.68e4T^{2} \)
11 \( 1 - 505. iT - 1.61e5T^{2} \)
13 \( 1 - 275.T + 3.71e5T^{2} \)
17 \( 1 - 1.48e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.10e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.69e3T + 6.43e6T^{2} \)
31 \( 1 - 1.90e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.56e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.83e4iT - 1.15e8T^{2} \)
43 \( 1 + 6.65e3iT - 1.47e8T^{2} \)
47 \( 1 + 6.49e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.38e4T + 4.18e8T^{2} \)
59 \( 1 + 3.02e4T + 7.14e8T^{2} \)
61 \( 1 - 1.70e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.21e4T + 1.35e9T^{2} \)
71 \( 1 + 4.56e4T + 1.80e9T^{2} \)
73 \( 1 + 5.63e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.64e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.55e4T + 3.93e9T^{2} \)
89 \( 1 + 5.78e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.08e5iT - 8.58e9T^{2} \)
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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

−16.09397746270081050650362435227, −15.27404406210398760452452383899, −15.05517911432791999738958652538, −12.34881972547266816498576541456, −11.19660678593841745796797683387, −10.41435776330599883031521101366, −8.664788276003764997320422789460, −7.06788002427095968059636191076, −4.70862553800798182214057082793, −3.39779508230200204247571400363, 0.852689832703470854248755877111, 3.10352875561412859168347171892, 6.24979559088998705734805716947, 7.47978228901518743423769133133, 8.218825903907106813499876158046, 11.23533308418729692052021785221, 11.73851440529779304689071545834, 12.85040427776589463212658962047, 14.32537811589158634199883336994, 15.80488364695650334301772141183

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