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Properties

Label 2-29-29.28-c9-0-21
Degree $2$
Conductor $29$
Sign $0.878 - 0.477i$
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  − 37.8i·2-s − 104. i·3-s − 924.·4-s − 2.68e3·5-s − 3.95e3·6-s + 4.79e3·7-s + 1.56e4i·8-s + 8.80e3·9-s + 1.01e5i·10-s − 5.35e4i·11-s + 9.64e4i·12-s − 6.59e4·13-s − 1.81e5i·14-s + 2.79e5i·15-s + 1.18e5·16-s − 9.11e4i·17-s + ⋯
L(s)  = 1  − 1.67i·2-s − 0.743i·3-s − 1.80·4-s − 1.91·5-s − 1.24·6-s + 0.754·7-s + 1.34i·8-s + 0.447·9-s + 3.21i·10-s − 1.10i·11-s + 1.34i·12-s − 0.640·13-s − 1.26i·14-s + 1.42i·15-s + 0.453·16-s − 0.264i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.878 - 0.477i$
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: \((2,\ 29,\ (\ :9/2),\ 0.878 - 0.477i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.241815 + 0.0615266i\)
\(L(\frac12)\) \(\approx\) \(0.241815 + 0.0615266i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.34e6 - 1.82e6i)T \)
good2 \( 1 + 37.8iT - 512T^{2} \)
3 \( 1 + 104. iT - 1.96e4T^{2} \)
5 \( 1 + 2.68e3T + 1.95e6T^{2} \)
7 \( 1 - 4.79e3T + 4.03e7T^{2} \)
11 \( 1 + 5.35e4iT - 2.35e9T^{2} \)
13 \( 1 + 6.59e4T + 1.06e10T^{2} \)
17 \( 1 + 9.11e4iT - 1.18e11T^{2} \)
19 \( 1 - 5.62e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.27e4T + 1.80e12T^{2} \)
31 \( 1 - 7.71e5iT - 2.64e13T^{2} \)
37 \( 1 - 1.39e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.80e7iT - 3.27e14T^{2} \)
43 \( 1 + 1.71e7iT - 5.02e14T^{2} \)
47 \( 1 + 4.69e7iT - 1.11e15T^{2} \)
53 \( 1 - 4.76e7T + 3.29e15T^{2} \)
59 \( 1 + 1.64e8T + 8.66e15T^{2} \)
61 \( 1 - 1.71e8iT - 1.16e16T^{2} \)
67 \( 1 + 3.13e8T + 2.72e16T^{2} \)
71 \( 1 + 1.66e8T + 4.58e16T^{2} \)
73 \( 1 + 2.67e7iT - 5.88e16T^{2} \)
79 \( 1 + 5.08e8iT - 1.19e17T^{2} \)
83 \( 1 + 4.73e7T + 1.86e17T^{2} \)
89 \( 1 + 8.92e8iT - 3.50e17T^{2} \)
97 \( 1 - 6.96e8iT - 7.60e17T^{2} \)
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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

−13.36538743717131965556717474341, −12.09320023754447191374735367295, −11.69386394893712539322029097528, −10.54144637202738326859296779693, −8.545704456281491085398878092583, −7.46940893097106760938832066293, −4.49078259681362931071445229219, −3.25938088243336061982153429539, −1.35789468238731837687535975816, −0.11305419820937986044903359207, 4.20527287546795970094859174734, 4.82795481323229723739559695903, 7.19811151846595821931868552502, 7.78570214993623837578765723005, 9.210361392714289267409518332672, 11.08524606773600038304899389851, 12.56964545839155875663757280569, 14.61032875546184023019956462645, 15.25824615533977122122763040776, 15.77568678703493981939504203672

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