Properties

Label 2-29-29.28-c5-0-2
Degree $2$
Conductor $29$
Sign $-0.891 + 0.453i$
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  + 9.47i·2-s + 19.9i·3-s − 57.7·4-s − 4.84·5-s − 188.·6-s + 219.·7-s − 243. i·8-s − 153.·9-s − 45.8i·10-s − 12.7i·11-s − 1.14e3i·12-s − 566.·13-s + 2.07e3i·14-s − 96.4i·15-s + 459.·16-s − 1.15e3i·17-s + ⋯
L(s)  = 1  + 1.67i·2-s + 1.27i·3-s − 1.80·4-s − 0.0866·5-s − 2.13·6-s + 1.69·7-s − 1.34i·8-s − 0.631·9-s − 0.145i·10-s − 0.0318i·11-s − 2.30i·12-s − 0.930·13-s + 2.83i·14-s − 0.110i·15-s + 0.448·16-s − 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.891 + 0.453i$
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: \((2,\ 29,\ (\ :5/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.342878 - 1.42830i\)
\(L(\frac12)\) \(\approx\) \(0.342878 - 1.42830i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-4.03e3 + 2.05e3i)T \)
good2 \( 1 - 9.47iT - 32T^{2} \)
3 \( 1 - 19.9iT - 243T^{2} \)
5 \( 1 + 4.84T + 3.12e3T^{2} \)
7 \( 1 - 219.T + 1.68e4T^{2} \)
11 \( 1 + 12.7iT - 1.61e5T^{2} \)
13 \( 1 + 566.T + 3.71e5T^{2} \)
17 \( 1 + 1.15e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.22e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.38e3T + 6.43e6T^{2} \)
31 \( 1 + 2.58e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.24e4iT - 6.93e7T^{2} \)
41 \( 1 - 3.68e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.28e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.91e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.19e4T + 4.18e8T^{2} \)
59 \( 1 - 6.94e3T + 7.14e8T^{2} \)
61 \( 1 + 2.48e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.39e4T + 1.35e9T^{2} \)
71 \( 1 + 1.20e4T + 1.80e9T^{2} \)
73 \( 1 - 3.94e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.46e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.19e4T + 3.93e9T^{2} \)
89 \( 1 + 6.43e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.43e5iT - 8.58e9T^{2} \)
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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

−16.66942453403642155899674399273, −15.39112008292972048878684720114, −14.84697788236024581173699656745, −13.91924553368267991045009589456, −11.57723091736498679889987885748, −9.962746207960671387682582580171, −8.574694341005088893323135581781, −7.42442622705390038099499834555, −5.30823304944515793088920272159, −4.51254702568705128808533473369, 1.08598303485468946932463304748, 2.31692293032993411950729489182, 4.73750556033271275606294268209, 7.38828058521619157974616708510, 8.817487474050067594132413165551, 10.71251873412272335235265523047, 11.65247411693095651504108638474, 12.58091907690069720621496944415, 13.59623721397368336068601909030, 14.82094561044363926157479526129

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