Properties

Label 2-29-29.28-c15-0-28
Degree $2$
Conductor $29$
Sign $-0.999 - 0.0402i$
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  − 202. i·2-s − 5.79e3i·3-s − 8.08e3·4-s + 1.72e5·5-s − 1.17e6·6-s + 3.88e6·7-s − 4.98e6i·8-s − 1.92e7·9-s − 3.49e7i·10-s + 1.04e8i·11-s + 4.68e7i·12-s + 2.38e8·13-s − 7.84e8i·14-s − 1.00e9i·15-s − 1.27e9·16-s − 8.65e8i·17-s + ⋯
L(s)  = 1  − 1.11i·2-s − 1.53i·3-s − 0.246·4-s + 0.990·5-s − 1.70·6-s + 1.78·7-s − 0.841i·8-s − 1.34·9-s − 1.10i·10-s + 1.61i·11-s + 0.377i·12-s + 1.05·13-s − 1.98i·14-s − 1.51i·15-s − 1.18·16-s − 0.511i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0402i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.999 - 0.0402i$
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: \((2,\ 29,\ (\ :15/2),\ -0.999 - 0.0402i)\)

Particular Values

\(L(8)\) \(\approx\) \(3.443999448\)
\(L(\frac12)\) \(\approx\) \(3.443999448\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (9.28e10 + 3.74e9i)T \)
good2 \( 1 + 202. iT - 3.27e4T^{2} \)
3 \( 1 + 5.79e3iT - 1.43e7T^{2} \)
5 \( 1 - 1.72e5T + 3.05e10T^{2} \)
7 \( 1 - 3.88e6T + 4.74e12T^{2} \)
11 \( 1 - 1.04e8iT - 4.17e15T^{2} \)
13 \( 1 - 2.38e8T + 5.11e16T^{2} \)
17 \( 1 + 8.65e8iT - 2.86e18T^{2} \)
19 \( 1 + 6.68e9iT - 1.51e19T^{2} \)
23 \( 1 - 7.33e9T + 2.66e20T^{2} \)
31 \( 1 + 6.84e10iT - 2.34e22T^{2} \)
37 \( 1 - 6.08e11iT - 3.33e23T^{2} \)
41 \( 1 - 6.76e11iT - 1.55e24T^{2} \)
43 \( 1 - 1.65e12iT - 3.17e24T^{2} \)
47 \( 1 - 2.89e11iT - 1.20e25T^{2} \)
53 \( 1 + 6.08e12T + 7.31e25T^{2} \)
59 \( 1 + 1.27e13T + 3.65e26T^{2} \)
61 \( 1 + 4.08e12iT - 6.02e26T^{2} \)
67 \( 1 - 7.48e13T + 2.46e27T^{2} \)
71 \( 1 + 1.13e13T + 5.87e27T^{2} \)
73 \( 1 + 3.81e13iT - 8.90e27T^{2} \)
79 \( 1 + 2.03e14iT - 2.91e28T^{2} \)
83 \( 1 - 3.95e14T + 6.11e28T^{2} \)
89 \( 1 - 3.87e14iT - 1.74e29T^{2} \)
97 \( 1 - 5.65e14iT - 6.33e29T^{2} \)
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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

−13.00471229054130868369369099279, −11.76902920007798514033267671284, −11.01055306274703125364706252342, −9.365922304158246828491826384135, −7.70577596348568182757459748977, −6.59697750535271366778923953694, −4.80342063692414165766250996221, −2.40176054832308182168878818792, −1.74178230897787689450523461727, −1.04318936719209723349566935428, 1.70132465680862720622381211301, 3.80303821054052043859305221027, 5.39364245072209187518549198454, 5.83466318612382892561757171097, 8.114586675699890923273918622551, 8.888754619019602456406610389843, 10.64620652098379762236323596598, 11.23348182234347432901289566684, 13.97548151793596847492109811225, 14.43966500430225560758903214535

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