LMFDB - L-function 2-29-29.28-c15-0-26

Dirichlet series

L(s) = 1

− 68.3i·2^-s − 2.97e3i·3^-s + 2.80e4·4^-s + 3.02e5·5^-s − 2.03e5·6^-s − 1.53e5·7^-s − 4.16e6i·8^-s + 5.50e6·9^-s − 2.06e7i·10^-s − 6.80e7i·11^-s − 8.35e7i·12^-s + 2.60e8·13^-s + 1.04e7i·14^-s − 8.99e8i·15^-s + 6.36e8·16^-s + 1.34e9i·17^-s + ⋯

L(s) = 1

− 0.377i·2^-s − 0.784i·3^-s + 0.857·4^-s + 1.73·5^-s − 0.296·6^-s − 0.0702·7^-s − 0.701i·8^-s + 0.383·9^-s − 0.654i·10^-s − 1.05i·11^-s − 0.672i·12^-s + 1.15·13^-s + 0.0265i·14^-s − 1.35i·15^-s + 0.592·16^-s + 0.792i·17^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(16-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$29$
Sign:	$0.227 + 0.973i$
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:	$(2,\ 29,\ (\ :15/2),\ 0.227 + 0.973i)$

Particular Values

$L(8)$	$\approx$	$4.040506753$
$L(\frac12)$	$\approx$	$4.040506753$
$L(\frac{17}{2})$		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 + (-2.11e10 - 9.04e10i)T $
good	2	$ 1 + 68.3iT - 3.27e4T^{2} $
	3	$ 1 + 2.97e3iT - 1.43e7T^{2} $
	5	$ 1 - 3.02e5T + 3.05e10T^{2} $
	7	$ 1 + 1.53e5T + 4.74e12T^{2} $
	11	$ 1 + 6.80e7iT - 4.17e15T^{2} $
	13	$ 1 - 2.60e8T + 5.11e16T^{2} $
	17	$ 1 - 1.34e9iT - 2.86e18T^{2} $
	19	$ 1 - 7.27e9iT - 1.51e19T^{2} $
	23	$ 1 + 9.77e9T + 2.66e20T^{2} $
	31	$ 1 + 4.75e10iT - 2.34e22T^{2} $
	37	$ 1 - 8.65e11iT - 3.33e23T^{2} $
	41	$ 1 + 7.42e11iT - 1.55e24T^{2} $
	43	$ 1 - 3.70e11iT - 3.17e24T^{2} $
	47	$ 1 + 7.31e11iT - 1.20e25T^{2} $
	53	$ 1 + 1.02e13T + 7.31e25T^{2} $
	59	$ 1 + 3.20e13T + 3.65e26T^{2} $
	61	$ 1 + 4.06e13iT - 6.02e26T^{2} $
	67	$ 1 - 1.43e13T + 2.46e27T^{2} $
	71	$ 1 + 5.74e13T + 5.87e27T^{2} $
	73	$ 1 - 1.16e13iT - 8.90e27T^{2} $
	79	$ 1 + 2.57e14iT - 2.91e28T^{2} $
	83	$ 1 + 4.36e14T + 6.11e28T^{2} $
	89	$ 1 - 6.21e14iT - 1.74e29T^{2} $
	97	$ 1 - 3.36e13iT - 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

−13.29442865172581471249335065969, −12.38376122018145357277834176519, −10.80789726591451183466909370114, −9.914417397401873690545280749659, −8.180189443699495382876897872029, −6.36699481813113820207712082928, −6.00013521683191012250523359603, −3.30666635256272918910881123282, −1.70314441614195848560957606918, −1.38284448121399971728961177075, 1.52261940718653181787281016362, 2.66332741953397711321619513854, 4.74827912868348344909304001773, 6.01495162776628532973355954054, 7.09014034536808854491863240935, 9.164353848514024964074715070062, 10.06406265306206981610173515004, 11.13324166990120114435797816101, 12.93847801962648920892062662269, 14.08488886712267155567492370944

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 $\F_{q}$

\(L(8)\)	\(\approx\)	\(4.040506753\)
\(L(\frac12)\)	\(\approx\)	\(4.040506753\)
\(L(\frac{17}{2})\)		not available
\(L(1)\)		not available

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