LMFDB - L-function 2-350-5.4-c9-0-40

Dirichlet series

L(s) = 1

+ 16i·2^-s − 247. i·3^-s − 256·4^-s + 3.95e3·6^-s − 2.40e3i·7^-s − 4.09e3i·8^-s − 4.13e4·9^-s − 2.79e4·11^-s + 6.32e4i·12^-s + 6.09e4i·13^-s + 3.84e4·14^-s + 6.55e4·16^-s + 3.58e5i·17^-s − 6.61e5i·18^-s + 3.91e5·19^-s + ⋯

L(s) = 1

+ 0.707i·2^-s − 1.76i·3^-s − 0.5·4^-s + 1.24·6^-s − 0.377i·7^-s − 0.353i·8^-s − 2.10·9^-s − 0.575·11^-s + 0.880i·12^-s + 0.591i·13^-s + 0.267·14^-s + 0.250·16^-s + 1.04i·17^-s − 1.48i·18^-s + 0.689·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$350$ = $2 \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$180.262$
Root analytic conductor:	$13.4261$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{350} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 350,\ (\ :9/2),\ 0.447 + 0.894i)$

Particular Values

$L(5)$	$\approx$	$1.544447583$
$L(\frac12)$	$\approx$	$1.544447583$
$L(\frac{11}{2})$		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 - 16iT $
	5	$ 1 $
	7	$ 1 + 2.40e3iT $
good	3	$ 1 + 247. iT - 1.96e4T^{2} $
	11	$ 1 + 2.79e4T + 2.35e9T^{2} $
	13	$ 1 - 6.09e4iT - 1.06e10T^{2} $
	17	$ 1 - 3.58e5iT - 1.18e11T^{2} $
	19	$ 1 - 3.91e5T + 3.22e11T^{2} $
	23	$ 1 + 3.02e5iT - 1.80e12T^{2} $
	29	$ 1 + 6.73e6T + 1.45e13T^{2} $
	31	$ 1 - 2.98e6T + 2.64e13T^{2} $
	37	$ 1 - 3.49e6iT - 1.29e14T^{2} $
	41	$ 1 - 3.43e7T + 3.27e14T^{2} $
	43	$ 1 + 1.45e7iT - 5.02e14T^{2} $
	47	$ 1 - 2.76e7iT - 1.11e15T^{2} $
	53	$ 1 + 2.39e7iT - 3.29e15T^{2} $
	59	$ 1 - 1.20e8T + 8.66e15T^{2} $
	61	$ 1 - 7.23e7T + 1.16e16T^{2} $
	67	$ 1 + 8.70e7iT - 2.72e16T^{2} $
	71	$ 1 - 2.19e8T + 4.58e16T^{2} $
	73	$ 1 - 2.67e8iT - 5.88e16T^{2} $
	79	$ 1 + 2.85e7T + 1.19e17T^{2} $
	83	$ 1 + 3.83e8iT - 1.86e17T^{2} $
	89	$ 1 + 7.21e8T + 3.50e17T^{2} $
	97	$ 1 - 6.73e8iT - 7.60e17T^{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

−9.506288561758611140894956816889, −8.404565181487977396050090251402, −7.71612519763611791672363360939, −7.05579807146087225785151045941, −6.20907223832616276183125021313, −5.39683455804801771614696291648, −3.89777308237504400049986968806, −2.47010369141997292147906833787, −1.43504500060080138980852634751, −0.47693266285094014636214173930, 0.60782990218019873339909391504, 2.43277149173576205614027645521, 3.21777988512367807790420084159, 4.13657470376384663088260517824, 5.16446045494403661167480536021, 5.66290001011185062589471096680, 7.58983904246545216699882258509, 8.705118762282516765806716239885, 9.548109677315911411030514303112, 9.970864032218899372911713486138

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.544447583\)
\(L(\frac12)\)	\(\approx\)	\(1.544447583\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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