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Properties

Label 2-29-29.28-c15-0-19
Degree $2$
Conductor $29$
Sign $0.839 - 0.543i$
Analytic cond. $41.3811$
Root an. cond. $6.43281$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 84.1i·2-s − 2.44e3i·3-s + 2.56e4·4-s + 2.24e4·5-s + 2.05e5·6-s + 3.47e6·7-s + 4.91e6i·8-s + 8.37e6·9-s + 1.88e6i·10-s + 5.52e7i·11-s − 6.27e7i·12-s − 4.72e7·13-s + 2.92e8i·14-s − 5.47e7i·15-s + 4.28e8·16-s − 5.10e8i·17-s + ⋯
L(s)  = 1  + 0.464i·2-s − 0.645i·3-s + 0.783·4-s + 0.128·5-s + 0.299·6-s + 1.59·7-s + 0.829i·8-s + 0.583·9-s + 0.0596i·10-s + 0.854i·11-s − 0.505i·12-s − 0.208·13-s + 0.740i·14-s − 0.0827i·15-s + 0.398·16-s − 0.301i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(41.3811\)
Root analytic conductor: \(6.43281\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :15/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(8)\) \(\approx\) \(3.416337022\)
\(L(\frac12)\) \(\approx\) \(3.416337022\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-7.79e10 + 5.04e10i)T \)
good2 \( 1 - 84.1iT - 3.27e4T^{2} \)
3 \( 1 + 2.44e3iT - 1.43e7T^{2} \)
5 \( 1 - 2.24e4T + 3.05e10T^{2} \)
7 \( 1 - 3.47e6T + 4.74e12T^{2} \)
11 \( 1 - 5.52e7iT - 4.17e15T^{2} \)
13 \( 1 + 4.72e7T + 5.11e16T^{2} \)
17 \( 1 + 5.10e8iT - 2.86e18T^{2} \)
19 \( 1 - 5.33e9iT - 1.51e19T^{2} \)
23 \( 1 + 6.90e9T + 2.66e20T^{2} \)
31 \( 1 - 8.33e9iT - 2.34e22T^{2} \)
37 \( 1 + 6.22e11iT - 3.33e23T^{2} \)
41 \( 1 - 1.08e12iT - 1.55e24T^{2} \)
43 \( 1 - 1.44e12iT - 3.17e24T^{2} \)
47 \( 1 - 6.00e12iT - 1.20e25T^{2} \)
53 \( 1 - 6.81e12T + 7.31e25T^{2} \)
59 \( 1 - 2.38e13T + 3.65e26T^{2} \)
61 \( 1 + 1.51e13iT - 6.02e26T^{2} \)
67 \( 1 + 6.27e13T + 2.46e27T^{2} \)
71 \( 1 - 7.71e13T + 5.87e27T^{2} \)
73 \( 1 + 9.88e13iT - 8.90e27T^{2} \)
79 \( 1 + 3.04e14iT - 2.91e28T^{2} \)
83 \( 1 + 1.95e14T + 6.11e28T^{2} \)
89 \( 1 + 3.35e14iT - 1.74e29T^{2} \)
97 \( 1 + 2.00e14iT - 6.33e29T^{2} \)
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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

−14.08334761987832759721228570353, −12.39680318524751732238903415639, −11.51949506479183224179660500420, −10.11641210180253829659399707762, −8.000024820388523080063335142517, −7.45095870046446319768473763057, −6.01741585437754803316874113340, −4.53407008424715605628547748892, −2.13851164343400654831158214001, −1.43165299537401054892354461232, 1.06271630876779729797293743494, 2.27525020689296341397015603251, 3.91806116552774604311288628050, 5.27400310353479924219344793442, 7.04822558238161213492772497783, 8.488418597098327352658054569732, 10.14441096374991490553825890074, 11.05333157789109446363484983147, 11.92772691791332956411993197760, 13.59938605086449993493005693210

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