LMFDB - L-function 2-48e2-24.11-c1-0-8

Dirichlet series

L(s) = 1

+ 1.41·5^-s − 4i·7^-s + 5.65i·11^-s + 4i·13^-s + 4.24i·17^-s − 5.65·23^-s − 2.99·25^-s + 1.41·29^-s + 4i·31^-s − 5.65i·35^-s + 6i·37^-s + 9.89i·41^-s − 8·43^-s + 5.65·47^-s − 9·49^-s + ⋯

L(s) = 1

+ 0.632·5^-s − 1.51i·7^-s + 1.70i·11^-s + 1.10i·13^-s + 1.02i·17^-s − 1.17·23^-s − 0.599·25^-s + 0.262·29^-s + 0.718i·31^-s − 0.956i·35^-s + 0.986i·37^-s + 1.54i·41^-s − 1.21·43^-s + 0.825·47^-s − 1.28·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.169 - 0.985i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ 0.169 - 0.985i)$

Particular Values

$L(1)$	$\approx$	$1.514968959$
$L(\frac12)$	$\approx$	$1.514968959$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
good	5	$ 1 - 1.41T + 5T^{2} $
	7	$ 1 + 4iT - 7T^{2} $
	11	$ 1 - 5.65iT - 11T^{2} $
	13	$ 1 - 4iT - 13T^{2} $
	17	$ 1 - 4.24iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 5.65T + 23T^{2} $
	29	$ 1 - 1.41T + 29T^{2} $
	31	$ 1 - 4iT - 31T^{2} $
	37	$ 1 - 6iT - 37T^{2} $
	41	$ 1 - 9.89iT - 41T^{2} $
	43	$ 1 + 8T + 43T^{2} $
	47	$ 1 - 5.65T + 47T^{2} $
	53	$ 1 - 4.24T + 53T^{2} $
	59	$ 1 + 11.3iT - 59T^{2} $
	61	$ 1 + 2iT - 61T^{2} $
	67	$ 1 - 8T + 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 + 4iT - 79T^{2} $
	83	$ 1 - 5.65iT - 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 + 8T + 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

−9.449149393075227989730789888973, −8.273026049076786164650852656243, −7.60506648164421777492938204802, −6.71426933951628523497791878928, −6.39847548058666292754529106067, −5.02248937889754393599984368976, −4.32567341229265526770809820855, −3.68915523240855936277330940246, −2.10138130112817854328994691391, −1.45586071321267429369065148834, 0.50094540756038180777404771886, 2.15023267609907549640070515740, 2.82542298681241372053169947846, 3.78086607665832519956693901093, 5.26197659851575260690678103060, 5.74475962828882353794556017984, 6.08737521746409157262190463036, 7.37247979145744516913647744045, 8.316123274921053911809437344698, 8.747046075935062679668767777218

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.514968959\)
\(L(\frac12)\)	\(\approx\)	\(1.514968959\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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