LMFDB - L-function 2-5040-21.20-c1-0-51

Dirichlet series

L(s) = 1

− 5^-s + (−0.0951 + 2.64i)7^-s − 5.28i·11^-s + 2.19i·13^-s + 1.04·17^-s − 6.43i·19^-s − 7.47i·23^-s + 25^-s + 7.47i·29^-s + 9.09i·31^-s + (0.0951 − 2.64i)35^-s − 0.855·37^-s − 2.19·41^-s + 0.954·43^-s − 11.0·47^-s + ⋯

L(s) = 1

− 0.447·5^-s + (−0.0359 + 0.999i)7^-s − 1.59i·11^-s + 0.607i·13^-s + 0.253·17^-s − 1.47i·19^-s − 1.55i·23^-s + 0.200·25^-s + 1.38i·29^-s + 1.63i·31^-s + (0.0160 − 0.446i)35^-s − 0.140·37^-s − 0.342·41^-s + 0.145·43^-s − 1.60·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5040$ = $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Sign:	$-0.836 + 0.547i$
Analytic conductor:	$40.2446$
Root analytic conductor:	$6.34386$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5040} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5040,\ (\ :1/2),\ -0.836 + 0.547i)$

Particular Values

$L(1)$	$\approx$	$0.4413896154$
$L(\frac12)$	$\approx$	$0.4413896154$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	7	$ 1 + (0.0951 - 2.64i)T $
good	11	$ 1 + 5.28iT - 11T^{2} $
	13	$ 1 - 2.19iT - 13T^{2} $
	17	$ 1 - 1.04T + 17T^{2} $
	19	$ 1 + 6.43iT - 19T^{2} $
	23	$ 1 + 7.47iT - 23T^{2} $
	29	$ 1 - 7.47iT - 29T^{2} $
	31	$ 1 - 9.09iT - 31T^{2} $
	37	$ 1 + 0.855T + 37T^{2} $
	41	$ 1 + 2.19T + 41T^{2} $
	43	$ 1 - 0.954T + 43T^{2} $
	47	$ 1 + 11.0T + 47T^{2} $
	53	$ 1 + 3.09iT - 53T^{2} $
	59	$ 1 - 13.7T + 59T^{2} $
	61	$ 1 - 8.05iT - 61T^{2} $
	67	$ 1 + 5.33T + 67T^{2} $
	71	$ 1 - 6.43iT - 71T^{2} $
	73	$ 1 - 4.57iT - 73T^{2} $
	79	$ 1 + 15.6T + 79T^{2} $
	83	$ 1 + 4.38T + 83T^{2} $
	89	$ 1 + 4.28T + 89T^{2} $
	97	$ 1 + 11.8iT - 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

−8.321488228754995380761214920814, −6.91296433261602343972345568310, −6.73587007114032601103835795234, −5.67010615748649237727142520222, −5.10080844863234680826946994031, −4.25685383281580170905843231439, −3.11524479789980278047727264052, −2.78419743312902339519311792915, −1.40504303783983409760650919522, −0.12319444385561396371680421981, 1.26720492413828452035832569673, 2.19266821540687892750539315141, 3.49324481666117236229477623632, 3.98370186592904489465328207127, 4.74018045702458173166755507005, 5.61928165390656916164256966551, 6.43480578340247466277690654954, 7.35803816104168685381480749773, 7.71996349946965925555746278875, 8.182217301881173607146699625663

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4413896154\)
\(L(\frac12)\)	\(\approx\)	\(0.4413896154\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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