LMFDB - L-function 2-29-29.28-c15-0-17

Dirichlet series

L(s) = 1

− 14.2i·2^-s + 3.94e3i·3^-s + 3.25e4·4^-s − 1.25e5·5^-s + 5.61e4·6^-s − 3.40e6·7^-s − 9.30e5i·8^-s − 1.20e6·9^-s + 1.78e6i·10^-s − 8.54e7i·11^-s + 1.28e8i·12^-s + 2.96e8·13^-s + 4.85e7i·14^-s − 4.94e8i·15^-s + 1.05e9·16^-s − 2.42e9i·17^-s + ⋯

L(s) = 1

− 0.0787i·2^-s + 1.04i·3^-s + 0.993·4^-s − 0.717·5^-s + 0.0819·6^-s − 1.56·7^-s − 0.156i·8^-s − 0.0841·9^-s + 0.0564i·10^-s − 1.32i·11^-s + 1.03i·12^-s + 1.30·13^-s + 0.123i·14^-s − 0.746i·15^-s + 0.981·16^-s − 1.43i·17^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(16-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$29$
Sign:	$0.983 - 0.180i$
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:	$(2,\ 29,\ (\ :15/2),\ 0.983 - 0.180i)$

Particular Values

$L(8)$	$\approx$	$2.019580820$
$L(\frac12)$	$\approx$	$2.019580820$
$L(\frac{17}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	29	$ 1 + (-9.13e10 + 1.68e10i)T $
good	2	$ 1 + 14.2iT - 3.27e4T^{2} $
	3	$ 1 - 3.94e3iT - 1.43e7T^{2} $
	5	$ 1 + 1.25e5T + 3.05e10T^{2} $
	7	$ 1 + 3.40e6T + 4.74e12T^{2} $
	11	$ 1 + 8.54e7iT - 4.17e15T^{2} $
	13	$ 1 - 2.96e8T + 5.11e16T^{2} $
	17	$ 1 + 2.42e9iT - 2.86e18T^{2} $
	19	$ 1 - 4.03e9iT - 1.51e19T^{2} $
	23	$ 1 - 1.05e10T + 2.66e20T^{2} $
	31	$ 1 - 2.30e11iT - 2.34e22T^{2} $
	37	$ 1 + 1.11e11iT - 3.33e23T^{2} $
	41	$ 1 + 1.39e12iT - 1.55e24T^{2} $
	43	$ 1 + 1.88e12iT - 3.17e24T^{2} $
	47	$ 1 - 3.19e12iT - 1.20e25T^{2} $
	53	$ 1 - 1.02e13T + 7.31e25T^{2} $
	59	$ 1 - 6.18e12T + 3.65e26T^{2} $
	61	$ 1 - 3.39e13iT - 6.02e26T^{2} $
	67	$ 1 - 7.36e13T + 2.46e27T^{2} $
	71	$ 1 - 7.54e13T + 5.87e27T^{2} $
	73	$ 1 + 1.04e14iT - 8.90e27T^{2} $
	79	$ 1 + 2.57e14iT - 2.91e28T^{2} $
	83	$ 1 + 5.96e13T + 6.11e28T^{2} $
	89	$ 1 - 2.57e14iT - 1.74e29T^{2} $
	97	$ 1 + 1.01e15iT - 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

−13.76704911532428696812369039398, −12.24104045790205292312798763802, −11.05445172739919942931570049080, −10.14300210945036961067756953329, −8.759591431547195164733388628944, −7.00662536606148053809921771523, −5.78679415577617943418258119953, −3.68086897337399526632402799111, −3.13947199467989615259297976319, −0.73424569112371983327847937648, 0.955827466652349927839110366671, 2.33195434015837780214520428177, 3.78128443980550900688285568609, 6.32191819211463268697737396639, 6.86895286715318342669208594409, 8.066829478590043978866419127168, 9.952221614982869755555428642664, 11.40171177021556309065071775737, 12.61372144112120926213515898752, 13.14167490846651707748095621874

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(8)\)	\(\approx\)	\(2.019580820\)
\(L(\frac12)\)	\(\approx\)	\(2.019580820\)
\(L(\frac{17}{2})\)		not available
\(L(1)\)		not available

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