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Properties

Label 2-29-29.28-c9-0-13
Degree $2$
Conductor $29$
Sign $0.300 + 0.953i$
Analytic cond. $14.9360$
Root an. cond. $3.86471$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.67i·2-s − 77.6i·3-s + 467.·4-s − 531.·5-s − 518.·6-s + 5.98e3·7-s − 6.53e3i·8-s + 1.36e4·9-s + 3.54e3i·10-s + 3.56e4i·11-s − 3.62e4i·12-s − 3.93e4·13-s − 3.99e4i·14-s + 4.12e4i·15-s + 1.95e5·16-s − 6.56e5i·17-s + ⋯
L(s)  = 1  − 0.294i·2-s − 0.553i·3-s + 0.913·4-s − 0.380·5-s − 0.163·6-s + 0.942·7-s − 0.564i·8-s + 0.693·9-s + 0.112i·10-s + 0.734i·11-s − 0.505i·12-s − 0.381·13-s − 0.277i·14-s + 0.210i·15-s + 0.746·16-s − 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.300 + 0.953i$
Analytic conductor: \(14.9360\)
Root analytic conductor: \(3.86471\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :9/2),\ 0.300 + 0.953i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.96438 - 1.43999i\)
\(L(\frac12)\) \(\approx\) \(1.96438 - 1.43999i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (1.14e6 + 3.63e6i)T \)
good2 \( 1 + 6.67iT - 512T^{2} \)
3 \( 1 + 77.6iT - 1.96e4T^{2} \)
5 \( 1 + 531.T + 1.95e6T^{2} \)
7 \( 1 - 5.98e3T + 4.03e7T^{2} \)
11 \( 1 - 3.56e4iT - 2.35e9T^{2} \)
13 \( 1 + 3.93e4T + 1.06e10T^{2} \)
17 \( 1 + 6.56e5iT - 1.18e11T^{2} \)
19 \( 1 + 3.67e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.94e6T + 1.80e12T^{2} \)
31 \( 1 - 6.74e5iT - 2.64e13T^{2} \)
37 \( 1 - 2.24e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.38e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.78e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.13e7iT - 1.11e15T^{2} \)
53 \( 1 - 7.62e7T + 3.29e15T^{2} \)
59 \( 1 + 1.45e6T + 8.66e15T^{2} \)
61 \( 1 - 6.58e7iT - 1.16e16T^{2} \)
67 \( 1 - 1.66e8T + 2.72e16T^{2} \)
71 \( 1 + 1.87e8T + 4.58e16T^{2} \)
73 \( 1 + 2.99e8iT - 5.88e16T^{2} \)
79 \( 1 - 3.49e8iT - 1.19e17T^{2} \)
83 \( 1 + 6.55e8T + 1.86e17T^{2} \)
89 \( 1 - 1.20e8iT - 3.50e17T^{2} \)
97 \( 1 - 6.52e8iT - 7.60e17T^{2} \)
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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

−14.97468058043744749314846733010, −13.36720332390937461800339079422, −12.01983489431031106740415176765, −11.31861087207431299885281638928, −9.751624266558769818359187111290, −7.67059490561438534136675331333, −6.94418594115320212915504389286, −4.73957724722501576436693895466, −2.53001153240106129022847517864, −1.11500606111757373936643931879, 1.66761280616425657295240861838, 3.79852039375615858633263549660, 5.54533512155623205090796992083, 7.25306296329749489396039224195, 8.469952341780965749217698631199, 10.44235164035591147975228070489, 11.26357875495807461884397909009, 12.66121451194746602735932245173, 14.59868926352458359135877717484, 15.25364985726364342245617697391

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