LMFDB - L-function 2-70e2-5.4-c1-0-41

Dirichlet series

L(s) = 1

− 2.41i·3^-s − 2.82·9^-s + 1.82·11^-s − 6.41i·13^-s + 3.58i·17^-s + 7.65·19^-s − 3.41i·23^-s − 0.414i·27^-s + 4.65·29^-s + 7.41·31^-s − 4.41i·33^-s − 0.585i·37^-s − 15.4·39^-s + 3.41·41^-s − 0.343i·43^-s + ⋯

L(s) = 1

− 1.39i·3^-s − 0.942·9^-s + 0.551·11^-s − 1.77i·13^-s + 0.869i·17^-s + 1.75·19^-s − 0.711i·23^-s − 0.0797i·27^-s + 0.864·29^-s + 1.33·31^-s − 0.768i·33^-s − 0.0963i·37^-s − 2.47·39^-s + 0.533·41^-s − 0.0523i·43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4900$ = $2^{2} \cdot 5^{2} \cdot 7^{2}$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$39.1266$
Root analytic conductor:	$6.25513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4900} (2549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4900,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.246852013$
$L(\frac12)$	$\approx$	$2.246852013$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
	7	$ 1 $
good	3	$ 1 + 2.41iT - 3T^{2} $
	11	$ 1 - 1.82T + 11T^{2} $
	13	$ 1 + 6.41iT - 13T^{2} $
	17	$ 1 - 3.58iT - 17T^{2} $
	19	$ 1 - 7.65T + 19T^{2} $
	23	$ 1 + 3.41iT - 23T^{2} $
	29	$ 1 - 4.65T + 29T^{2} $
	31	$ 1 - 7.41T + 31T^{2} $
	37	$ 1 + 0.585iT - 37T^{2} $
	41	$ 1 - 3.41T + 41T^{2} $
	43	$ 1 + 0.343iT - 43T^{2} $
	47	$ 1 - 10.8iT - 47T^{2} $
	53	$ 1 - 12.2iT - 53T^{2} $
	59	$ 1 + 0.585T + 59T^{2} $
	61	$ 1 - 10.8T + 61T^{2} $
	67	$ 1 - 3.07iT - 67T^{2} $
	71	$ 1 + 10.4T + 71T^{2} $
	73	$ 1 + 10.8iT - 73T^{2} $
	79	$ 1 - 15.1T + 79T^{2} $
	83	$ 1 - 8iT - 83T^{2} $
	89	$ 1 - 16.9T + 89T^{2} $
	97	$ 1 + 9.72iT - 97T^{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

−7.85702214714817801290571005471, −7.49464457244106326482292869278, −6.52759582392083547785143883541, −6.07174600949835282999908399347, −5.31841161114391276987868350994, −4.30920553880049508166103750721, −3.14152472128562187555756998001, −2.59767516013113999166936826792, −1.28603940258140992574964638892, −0.78296298390531300076142340889, 1.10568913833045997911908541267, 2.39922619072475170118295870180, 3.47227237006939469496673212468, 3.95756469741968028386452128194, 4.90873737227662983601825081132, 5.16737017349073616703701956798, 6.38187474100163801327050614688, 6.95941054499430448311530361611, 7.80682736677298438023937372796, 8.893484449005863514537185191357

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(2.246852013\)
\(L(\frac12)\)	\(\approx\)	\(2.246852013\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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