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Properties

Label 2-29-29.28-c5-0-8
Degree $2$
Conductor $29$
Sign $-0.615 + 0.788i$
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  − 4.44i·2-s − 21.5i·3-s + 12.2·4-s + 90.0·5-s − 95.6·6-s − 211.·7-s − 196. i·8-s − 219.·9-s − 400. i·10-s + 593. i·11-s − 262. i·12-s + 445.·13-s + 941. i·14-s − 1.93e3i·15-s − 484.·16-s + 368. i·17-s + ⋯
L(s)  = 1  − 0.786i·2-s − 1.37i·3-s + 0.381·4-s + 1.61·5-s − 1.08·6-s − 1.63·7-s − 1.08i·8-s − 0.903·9-s − 1.26i·10-s + 1.47i·11-s − 0.526i·12-s + 0.731·13-s + 1.28i·14-s − 2.22i·15-s − 0.472·16-s + 0.309i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.615 + 0.788i$
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: \((2,\ 29,\ (\ :5/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.823805 - 1.68874i\)
\(L(\frac12)\) \(\approx\) \(0.823805 - 1.68874i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-2.78e3 + 3.56e3i)T \)
good2 \( 1 + 4.44iT - 32T^{2} \)
3 \( 1 + 21.5iT - 243T^{2} \)
5 \( 1 - 90.0T + 3.12e3T^{2} \)
7 \( 1 + 211.T + 1.68e4T^{2} \)
11 \( 1 - 593. iT - 1.61e5T^{2} \)
13 \( 1 - 445.T + 3.71e5T^{2} \)
17 \( 1 - 368. iT - 1.41e6T^{2} \)
19 \( 1 + 761. iT - 2.47e6T^{2} \)
23 \( 1 - 728.T + 6.43e6T^{2} \)
31 \( 1 - 9.22e3iT - 2.86e7T^{2} \)
37 \( 1 - 3.39e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.76e3iT - 1.15e8T^{2} \)
43 \( 1 + 5.29e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.24e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.47e4T + 4.18e8T^{2} \)
59 \( 1 + 3.45e4T + 7.14e8T^{2} \)
61 \( 1 - 9.45e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.10e4T + 1.35e9T^{2} \)
71 \( 1 + 2.34e4T + 1.80e9T^{2} \)
73 \( 1 + 2.54e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.54e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.34e4T + 3.93e9T^{2} \)
89 \( 1 - 1.12e5iT - 5.58e9T^{2} \)
97 \( 1 - 724. iT - 8.58e9T^{2} \)
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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

−15.72139749597699505279757725293, −13.72289166940339052221794384351, −12.88890738322936663299999617512, −12.36153075787333721709537046500, −10.36965727052532172907114388175, −9.453193122381445866928571809004, −6.88985254576572870954169668772, −6.29765994176217081785641672624, −2.67247057886995639797592110749, −1.44218048730810977177326815177, 3.12536577661870486281674587653, 5.66022465744797012013011346410, 6.34618940409405022036135927116, 8.912189184185007413097529752887, 9.904115153921783090276103990657, 10.95441966298128079044786525423, 13.24090716038807895396418262585, 14.26715548936483939606312309153, 15.69957378653108166745049319257, 16.38849509689627632129560130462

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