LMFDB - L-function 2-825-5.4-c3-0-48

Dirichlet series

L(s) = 1

− 2.32i·2^-s + 3i·3^-s + 2.57·4^-s + 6.98·6^-s + 22.4i·7^-s − 24.6i·8^-s − 9·9^-s + 11·11^-s + 7.72i·12^-s + 9.86i·13^-s + 52.3·14^-s − 36.7·16^-s − 128. i·17^-s + 20.9i·18^-s − 7.04·19^-s + ⋯

L(s) = 1

− 0.823i·2^-s + 0.577i·3^-s + 0.321·4^-s + 0.475·6^-s + 1.21i·7^-s − 1.08i·8^-s − 0.333·9^-s + 0.301·11^-s + 0.185i·12^-s + 0.210i·13^-s + 0.998·14^-s − 0.574·16^-s − 1.82i·17^-s + 0.274i·18^-s − 0.0850·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.503804898$
$L(\frac12)$	$\approx$	$2.503804898$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - 3iT $
	5	$ 1 $
	11	$ 1 - 11T $
good	2	$ 1 + 2.32iT - 8T^{2} $
	7	$ 1 - 22.4iT - 343T^{2} $
	13	$ 1 - 9.86iT - 2.19e3T^{2} $
	17	$ 1 + 128. iT - 4.91e3T^{2} $
	19	$ 1 + 7.04T + 6.85e3T^{2} $
	23	$ 1 + 0.654iT - 1.21e4T^{2} $
	29	$ 1 - 229.T + 2.43e4T^{2} $
	31	$ 1 - 155.T + 2.97e4T^{2} $
	37	$ 1 + 110. iT - 5.06e4T^{2} $
	41	$ 1 - 154.T + 6.89e4T^{2} $
	43	$ 1 - 401. iT - 7.95e4T^{2} $
	47	$ 1 + 277. iT - 1.03e5T^{2} $
	53	$ 1 - 651. iT - 1.48e5T^{2} $
	59	$ 1 - 423.T + 2.05e5T^{2} $
	61	$ 1 - 681.T + 2.26e5T^{2} $
	67	$ 1 - 374. iT - 3.00e5T^{2} $
	71	$ 1 - 96.6T + 3.57e5T^{2} $
	73	$ 1 - 19.9iT - 3.89e5T^{2} $
	79	$ 1 + 24.4T + 4.93e5T^{2} $
	83	$ 1 - 1.12e3iT - 5.71e5T^{2} $
	89	$ 1 - 639.T + 7.04e5T^{2} $
	97	$ 1 + 730. iT - 9.12e5T^{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

−9.750319283841036724399537705096, −9.269246554923196056201918990990, −8.345244918798560689392773833064, −7.09491414282375677104349427475, −6.23228579031414765530694019736, −5.19432293809100933391532477086, −4.20726825036438121215678725716, −2.90332956695719562614345734434, −2.44245855285065307557771572600, −0.880889683012004442283545164812, 0.907785664779031561846166386016, 2.09175622440596685539037590006, 3.50637291121786958191370652154, 4.63157648317143216868314167908, 5.87711681460189615587731021864, 6.57116644189669433945597803919, 7.17119465525150624955749983043, 8.121750292408667812640598002593, 8.519935369643459810724310302886, 10.08515833848656891405205724112

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.503804898\)
\(L(\frac12)\)	\(\approx\)	\(2.503804898\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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