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Properties

Label 2-29-29.28-c13-0-3
Degree $2$
Conductor $29$
Sign $0.903 - 0.428i$
Analytic cond. $31.0969$
Root an. cond. $5.57646$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 170. i·2-s + 2.48e3i·3-s − 2.08e4·4-s − 1.46e4·5-s − 4.23e5·6-s + 3.77e5·7-s − 2.15e6i·8-s − 4.59e6·9-s − 2.49e6i·10-s + 5.90e6i·11-s − 5.17e7i·12-s + 1.59e7·13-s + 6.43e7i·14-s − 3.64e7i·15-s + 1.95e8·16-s − 4.16e7i·17-s + ⋯
L(s)  = 1  + 1.88i·2-s + 1.96i·3-s − 2.54·4-s − 0.419·5-s − 3.70·6-s + 1.21·7-s − 2.90i·8-s − 2.87·9-s − 0.789i·10-s + 1.00i·11-s − 5.00i·12-s + 0.915·13-s + 2.28i·14-s − 0.826i·15-s + 2.91·16-s − 0.418i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.903 - 0.428i$
Analytic conductor: \(31.0969\)
Root analytic conductor: \(5.57646\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :13/2),\ 0.903 - 0.428i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.5836800524\)
\(L(\frac12)\) \(\approx\) \(0.5836800524\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.89e9 - 1.37e9i)T \)
good2 \( 1 - 170. iT - 8.19e3T^{2} \)
3 \( 1 - 2.48e3iT - 1.59e6T^{2} \)
5 \( 1 + 1.46e4T + 1.22e9T^{2} \)
7 \( 1 - 3.77e5T + 9.68e10T^{2} \)
11 \( 1 - 5.90e6iT - 3.45e13T^{2} \)
13 \( 1 - 1.59e7T + 3.02e14T^{2} \)
17 \( 1 + 4.16e7iT - 9.90e15T^{2} \)
19 \( 1 + 6.36e7iT - 4.20e16T^{2} \)
23 \( 1 + 5.72e8T + 5.04e17T^{2} \)
31 \( 1 + 4.15e9iT - 2.44e19T^{2} \)
37 \( 1 - 1.13e10iT - 2.43e20T^{2} \)
41 \( 1 + 1.59e10iT - 9.25e20T^{2} \)
43 \( 1 - 4.21e10iT - 1.71e21T^{2} \)
47 \( 1 - 1.43e10iT - 5.46e21T^{2} \)
53 \( 1 - 9.71e10T + 2.60e22T^{2} \)
59 \( 1 + 3.40e11T + 1.04e23T^{2} \)
61 \( 1 - 6.72e11iT - 1.61e23T^{2} \)
67 \( 1 - 8.29e11T + 5.48e23T^{2} \)
71 \( 1 + 4.77e11T + 1.16e24T^{2} \)
73 \( 1 + 9.83e11iT - 1.67e24T^{2} \)
79 \( 1 - 1.29e12iT - 4.66e24T^{2} \)
83 \( 1 + 5.00e12T + 8.87e24T^{2} \)
89 \( 1 + 2.93e12iT - 2.19e25T^{2} \)
97 \( 1 + 7.14e12iT - 6.73e25T^{2} \)
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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

−15.39509551049643157530475021283, −14.85961475993692475552316511097, −13.88713839792960961344839709522, −11.39106663092224622974789924841, −9.870175968262269503994213779553, −8.781790206549213657853483630140, −7.78079009888501566884356860618, −5.76885484814732540885314426361, −4.70416890674548791357052069803, −3.99545888639234134766175630988, 0.19986153512977894107886697331, 1.28997500353193071958826627850, 2.05632885975109262790181315131, 3.58670205690825168262671815703, 5.70456321358476716479929278501, 7.969836232479423422646080244164, 8.615373103395695082427878566637, 11.02907498984070942080085520484, 11.55686169073410661656692618713, 12.51190079759697391726229648954

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