LMFDB - L-function 2-2376-33.32-c1-0-24

Dirichlet series

L(s) = 1

− 2.21i·5^-s + 3.17i·7^-s + (1.18 − 3.09i)11^-s + 3.59i·13^-s − 2.88·17^-s − 1.52i·19^-s − 2.55i·23^-s + 0.114·25^-s + 6.60·29^-s − 0.572·31^-s + 7.00·35^-s + 9.30·37^-s + 5.01·41^-s − 1.96i·43^-s − 1.41i·47^-s + ⋯

L(s) = 1

− 0.988i·5^-s + 1.19i·7^-s + (0.357 − 0.933i)11^-s + 0.998i·13^-s − 0.699·17^-s − 0.349i·19^-s − 0.532i·23^-s + 0.0228·25^-s + 1.22·29^-s − 0.102·31^-s + 1.18·35^-s + 1.52·37^-s + 0.782·41^-s − 0.299i·43^-s − 0.206i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2376$ = $2^{3} \cdot 3^{3} \cdot 11$
Sign:	$0.933 + 0.357i$
Analytic conductor:	$18.9724$
Root analytic conductor:	$4.35573$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2376} (593, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2376,\ (\ :1/2),\ 0.933 + 0.357i)$

Particular Values

$L(1)$	$\approx$	$1.790696581$
$L(\frac12)$	$\approx$	$1.790696581$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-1.18 + 3.09i)T $
good	5	$ 1 + 2.21iT - 5T^{2} $
	7	$ 1 - 3.17iT - 7T^{2} $
	13	$ 1 - 3.59iT - 13T^{2} $
	17	$ 1 + 2.88T + 17T^{2} $
	19	$ 1 + 1.52iT - 19T^{2} $
	23	$ 1 + 2.55iT - 23T^{2} $
	29	$ 1 - 6.60T + 29T^{2} $
	31	$ 1 + 0.572T + 31T^{2} $
	37	$ 1 - 9.30T + 37T^{2} $
	41	$ 1 - 5.01T + 41T^{2} $
	43	$ 1 + 1.96iT - 43T^{2} $
	47	$ 1 + 1.41iT - 47T^{2} $
	53	$ 1 - 8.73iT - 53T^{2} $
	59	$ 1 - 4.64iT - 59T^{2} $
	61	$ 1 - 4.98iT - 61T^{2} $
	67	$ 1 + 11.0T + 67T^{2} $
	71	$ 1 + 9.37iT - 71T^{2} $
	73	$ 1 + 12.6iT - 73T^{2} $
	79	$ 1 + 0.575iT - 79T^{2} $
	83	$ 1 - 9.08T + 83T^{2} $
	89	$ 1 + 5.63iT - 89T^{2} $
	97	$ 1 - 18.1T + 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

−8.996642417006797745200754736567, −8.502986490568303155872010337838, −7.48140349631591373359412411959, −6.29354506326764667398887192675, −5.97672705952442911716995785244, −4.81017575503300045094910837313, −4.37071652687128652533463883982, −3.01813616186823704115735912811, −2.09585240292843303392422669543, −0.822003374206920997606497127615, 0.933195888491899673909347880015, 2.32575933797921318058608728054, 3.28507798826908020942957672221, 4.13625083061701843782234574037, 4.91442256104167054507400700027, 6.15192698035513143288431029035, 6.75689975326064316799562593553, 7.46683947521104331233994815366, 7.975077534321994219361945793240, 9.111136128333639816618172966912

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

\(L(1)\)	\(\approx\)	\(1.790696581\)
\(L(\frac12)\)	\(\approx\)	\(1.790696581\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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