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Properties

Label 2-29-29.28-c9-0-3
Degree $2$
Conductor $29$
Sign $-0.239 - 0.970i$
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  + 10.3i·2-s − 137. i·3-s + 404.·4-s − 2.02e3·5-s + 1.42e3·6-s − 4.87e3·7-s + 9.50e3i·8-s + 681.·9-s − 2.10e4i·10-s + 6.08e4i·11-s − 5.57e4i·12-s + 9.83e4·13-s − 5.05e4i·14-s + 2.79e5i·15-s + 1.08e5·16-s + 1.23e5i·17-s + ⋯
L(s)  = 1  + 0.458i·2-s − 0.982i·3-s + 0.790·4-s − 1.45·5-s + 0.450·6-s − 0.766·7-s + 0.820i·8-s + 0.0346·9-s − 0.665i·10-s + 1.25i·11-s − 0.776i·12-s + 0.955·13-s − 0.351i·14-s + 1.42i·15-s + 0.414·16-s + 0.358i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.239 - 0.970i$
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: \((2,\ 29,\ (\ :9/2),\ -0.239 - 0.970i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.657266 + 0.839207i\)
\(L(\frac12)\) \(\approx\) \(0.657266 + 0.839207i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-9.12e5 - 3.69e6i)T \)
good2 \( 1 - 10.3iT - 512T^{2} \)
3 \( 1 + 137. iT - 1.96e4T^{2} \)
5 \( 1 + 2.02e3T + 1.95e6T^{2} \)
7 \( 1 + 4.87e3T + 4.03e7T^{2} \)
11 \( 1 - 6.08e4iT - 2.35e9T^{2} \)
13 \( 1 - 9.83e4T + 1.06e10T^{2} \)
17 \( 1 - 1.23e5iT - 1.18e11T^{2} \)
19 \( 1 - 8.66e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.62e6T + 1.80e12T^{2} \)
31 \( 1 - 5.40e6iT - 2.64e13T^{2} \)
37 \( 1 + 4.18e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.65e7iT - 3.27e14T^{2} \)
43 \( 1 + 1.96e7iT - 5.02e14T^{2} \)
47 \( 1 - 5.21e7iT - 1.11e15T^{2} \)
53 \( 1 + 6.82e7T + 3.29e15T^{2} \)
59 \( 1 - 5.16e7T + 8.66e15T^{2} \)
61 \( 1 - 7.24e7iT - 1.16e16T^{2} \)
67 \( 1 - 2.50e6T + 2.72e16T^{2} \)
71 \( 1 + 1.90e7T + 4.58e16T^{2} \)
73 \( 1 + 1.51e8iT - 5.88e16T^{2} \)
79 \( 1 - 7.77e7iT - 1.19e17T^{2} \)
83 \( 1 - 4.38e8T + 1.86e17T^{2} \)
89 \( 1 + 2.50e8iT - 3.50e17T^{2} \)
97 \( 1 - 3.98e8iT - 7.60e17T^{2} \)
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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

−15.76852084842057735013259553214, −14.34850752213542645457175264457, −12.45729363090659438394577782105, −12.11426368892242139302144035478, −10.45094722030218126228509656979, −8.126744975584000379936902965772, −7.32096101827606065517616945926, −6.26359233740653010956299723010, −3.76759051853017392999102338904, −1.69449994818307657227869490275, 0.42306844250922282002917708028, 3.16607078254082189844689454425, 4.07536555608408306741353425199, 6.39808268985161491076699761708, 8.068401118391493178773041690464, 9.742003348494827343218521126584, 11.07916086203990520205647438427, 11.66863148266028044018102397304, 13.26599183394267782523153161781, 15.26724351504461596242285998545

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