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Properties

Label 2-29-29.28-c15-0-11
Degree $2$
Conductor $29$
Sign $-0.314 + 0.949i$
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  + 327. i·2-s + 2.16e3i·3-s − 7.42e4·4-s + 2.35e5·5-s − 7.07e5·6-s + 2.51e6·7-s − 1.35e7i·8-s + 9.66e6·9-s + 7.69e7i·10-s + 7.17e7i·11-s − 1.60e8i·12-s − 4.51e8·13-s + 8.22e8i·14-s + 5.08e8i·15-s + 2.00e9·16-s − 1.32e9i·17-s + ⋯
L(s)  = 1  + 1.80i·2-s + 0.571i·3-s − 2.26·4-s + 1.34·5-s − 1.03·6-s + 1.15·7-s − 2.28i·8-s + 0.673·9-s + 2.43i·10-s + 1.10i·11-s − 1.29i·12-s − 1.99·13-s + 2.08i·14-s + 0.768i·15-s + 1.87·16-s − 0.785i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.314 + 0.949i$
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: \((2,\ 29,\ (\ :15/2),\ -0.314 + 0.949i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.884359946\)
\(L(\frac12)\) \(\approx\) \(1.884359946\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.92e10 - 8.81e10i)T \)
good2 \( 1 - 327. iT - 3.27e4T^{2} \)
3 \( 1 - 2.16e3iT - 1.43e7T^{2} \)
5 \( 1 - 2.35e5T + 3.05e10T^{2} \)
7 \( 1 - 2.51e6T + 4.74e12T^{2} \)
11 \( 1 - 7.17e7iT - 4.17e15T^{2} \)
13 \( 1 + 4.51e8T + 5.11e16T^{2} \)
17 \( 1 + 1.32e9iT - 2.86e18T^{2} \)
19 \( 1 + 3.49e8iT - 1.51e19T^{2} \)
23 \( 1 + 2.49e10T + 2.66e20T^{2} \)
31 \( 1 - 2.04e11iT - 2.34e22T^{2} \)
37 \( 1 - 7.75e11iT - 3.33e23T^{2} \)
41 \( 1 - 9.31e11iT - 1.55e24T^{2} \)
43 \( 1 + 1.23e12iT - 3.17e24T^{2} \)
47 \( 1 - 2.93e12iT - 1.20e25T^{2} \)
53 \( 1 + 8.73e12T + 7.31e25T^{2} \)
59 \( 1 + 2.99e12T + 3.65e26T^{2} \)
61 \( 1 + 4.29e13iT - 6.02e26T^{2} \)
67 \( 1 - 5.77e13T + 2.46e27T^{2} \)
71 \( 1 - 4.14e13T + 5.87e27T^{2} \)
73 \( 1 + 5.77e12iT - 8.90e27T^{2} \)
79 \( 1 + 5.65e13iT - 2.91e28T^{2} \)
83 \( 1 - 1.19e14T + 6.11e28T^{2} \)
89 \( 1 - 6.03e14iT - 1.74e29T^{2} \)
97 \( 1 + 5.20e14iT - 6.33e29T^{2} \)
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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

−14.58316678311800026718164134781, −14.08139420889549567654949831185, −12.53267852216182404080309084538, −10.00803412832573341104872286318, −9.449769446736027563894166891049, −7.76571162476065576095196614009, −6.76297170404031771865453604505, −4.99542832924302982761556188472, −4.83393233869119836306000422728, −1.84676181923567372168443271966, 0.50330168349418980083528628344, 1.87458491782894673661563329048, 2.22920734685064976602680723999, 4.25833775402519112583671153326, 5.67006228418488433548631633393, 7.952050947656305646120400613290, 9.544301529092587375880740648339, 10.33581428882674016944586652392, 11.63367117653601963795886291322, 12.66524697804003705930951895957

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