LMFDB - L-function 2-48e2-24.11-c1-0-25

Dirichlet series

L(s) = 1

+ 1.41·5^-s − 4i·13^-s − 7.07i·17^-s − 2.99·25^-s − 9.89·29^-s − 2i·37^-s − 1.41i·41^-s + 7·49^-s − 7.07·53^-s − 10i·61^-s − 5.65i·65^-s + 16·73^-s − 10.0i·85^-s − 18.3i·89^-s + 8·97^-s + ⋯

L(s) = 1

+ 0.632·5^-s − 1.10i·13^-s − 1.71i·17^-s − 0.599·25^-s − 1.83·29^-s − 0.328i·37^-s − 0.220i·41^-s + 49^-s − 0.971·53^-s − 1.28i·61^-s − 0.701i·65^-s + 1.87·73^-s − 1.08i·85^-s − 1.94i·89^-s + 0.812·97^-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

$L(1)$	$\approx$	$1.460374532$
$L(\frac12)$	$\approx$	$1.460374532$
$L(\frac{3}{2})$		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 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 + 7.07iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 9.89T + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 + 2iT - 37T^{2} $
	41	$ 1 + 1.41iT - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 + 7.07T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 10iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 16T + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 18.3iT - 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

−8.993418330541367155686635936782, −7.85526641456180298664957157929, −7.39209145895184383829842853177, −6.39405722775993069261389145295, −5.52852446837176361142251981239, −5.05706155587407066680824631080, −3.81537170170190515161355602190, −2.87774810266515916159291507782, −1.92982315531864232176050593657, −0.47522688142941283647701939927, 1.53597012864433599737465886393, 2.23142613583721184599537317960, 3.63603980007675880511557931445, 4.27991131732866162932080464526, 5.44332080874189955729601884655, 6.07461015340323733131316569440, 6.79459440563583825221590381375, 7.71460175737035874011095293407, 8.506116605044931009084352098220, 9.304900188353494977524189979430

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.460374532\)
\(L(\frac12)\)	\(\approx\)	\(1.460374532\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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