LMFDB - L-function 2-72e2-8.5-c1-0-73

Dirichlet series

L(s) = 1

− 0.550i·11^-s − 7.89·17^-s + 6.34i·19^-s + 5·25^-s + 6.79·41^-s − 12.3i·43^-s − 7·49^-s − 15.2i·59^-s − 0.348i·67^-s − 15.6·73^-s − 18i·83^-s − 18·89^-s + 9.69·97^-s + 14.1i·107^-s − 18·113^-s + ⋯

L(s) = 1

− 0.165i·11^-s − 1.91·17^-s + 1.45i·19^-s + 25^-s + 1.06·41^-s − 1.88i·43^-s − 49^-s − 1.98i·59^-s − 0.0425i·67^-s − 1.83·73^-s − 1.97i·83^-s − 1.90·89^-s + 0.984·97^-s + 1.36i·107^-s − 1.69·113^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5184$ = $2^{6} \cdot 3^{4}$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$41.3944$
Root analytic conductor:	$6.43385$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5184} (2593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5184,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.5730105229$
$L(\frac12)$	$\approx$	$0.5730105229$
$L(\frac{3}{2})$		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 - 5T^{2} $
	7	$ 1 + 7T^{2} $
	11	$ 1 + 0.550iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 + 7.89T + 17T^{2} $
	19	$ 1 - 6.34iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 6.79T + 41T^{2} $
	43	$ 1 + 12.3iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 15.2iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 0.348iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 15.6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 18iT - 83T^{2} $
	89	$ 1 + 18T + 89T^{2} $
	97	$ 1 - 9.69T + 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

−8.000402256843471819270863838911, −7.17316526905583096747120796291, −6.51092367400465491359152894430, −5.85617180045887459271038890917, −4.96150058220779005019607860610, −4.23572679541714320769447782726, −3.45014835976629893229285510018, −2.43755126054069005645159908713, −1.58272822554165762556610007985, −0.15406605419796194453549698938, 1.18156922366678083546272482981, 2.42879719951160055302093179002, 2.95954224386891234670455741025, 4.36282698737847416310363769215, 4.55898799840352488369785733405, 5.58121939675583743691794379928, 6.53163250716911532934805661224, 6.90432129640558070559210043101, 7.72097218634256355160286605927, 8.647323874194498379470829570480

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5730105229\)
\(L(\frac12)\)	\(\approx\)	\(0.5730105229\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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