LMFDB - L-function 2-350-5.4-c5-0-25

Dirichlet series

L(s) = 1

− 4i·2^-s + 8i·3^-s − 16·4^-s + 32·6^-s + 49i·7^-s + 64i·8^-s + 179·9^-s − 340·11^-s − 128i·12^-s − 294i·13^-s + 196·14^-s + 256·16^-s − 1.22e3i·17^-s − 716i·18^-s − 2.43e3·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 0.513i·3^-s − 0.5·4^-s + 0.362·6^-s + 0.377i·7^-s + 0.353i·8^-s + 0.736·9^-s − 0.847·11^-s − 0.256i·12^-s − 0.482i·13^-s + 0.267·14^-s + 0.250·16^-s − 1.02i·17^-s − 0.520i·18^-s − 1.54·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$350$ = $2 \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$56.1343$
Root analytic conductor:	$7.49228$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{350} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 350,\ (\ :5/2),\ 0.447 + 0.894i)$

Particular Values

$L(3)$	$\approx$	$1.687995545$
$L(\frac12)$	$\approx$	$1.687995545$
$L(\frac{7}{2})$		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 + 4iT $
	5	$ 1 $
	7	$ 1 - 49iT $
good	3	$ 1 - 8iT - 243T^{2} $
	11	$ 1 + 340T + 1.61e5T^{2} $
	13	$ 1 + 294iT - 3.71e5T^{2} $
	17	$ 1 + 1.22e3iT - 1.41e6T^{2} $
	19	$ 1 + 2.43e3T + 2.47e6T^{2} $
	23	$ 1 - 2.00e3iT - 6.43e6T^{2} $
	29	$ 1 - 6.74e3T + 2.05e7T^{2} $
	31	$ 1 - 8.85e3T + 2.86e7T^{2} $
	37	$ 1 + 9.18e3iT - 6.93e7T^{2} $
	41	$ 1 + 1.45e4T + 1.15e8T^{2} $
	43	$ 1 - 8.10e3iT - 1.47e8T^{2} $
	47	$ 1 - 312iT - 2.29e8T^{2} $
	53	$ 1 + 1.46e4iT - 4.18e8T^{2} $
	59	$ 1 - 2.76e4T + 7.14e8T^{2} $
	61	$ 1 - 3.43e4T + 8.44e8T^{2} $
	67	$ 1 + 1.23e4iT - 1.35e9T^{2} $
	71	$ 1 - 3.69e4T + 1.80e9T^{2} $
	73	$ 1 + 6.17e4iT - 2.07e9T^{2} $
	79	$ 1 - 6.47e4T + 3.07e9T^{2} $
	83	$ 1 + 7.70e4iT - 3.93e9T^{2} $
	89	$ 1 - 8.16e3T + 5.58e9T^{2} $
	97	$ 1 + 2.06e4iT - 8.58e9T^{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

−10.33486471105591164635077800631, −9.915799826425612805473910191424, −8.794215056352537391386176721903, −7.909458893826201233397615466203, −6.60645263205309433371037271602, −5.21261292682522882761136799951, −4.48260092702335231231270061977, −3.20337790692895664389565678350, −2.13468701313522578176703887711, −0.56353517053865229587913572201, 0.897097503485893884329178376324, 2.31324394472296144315754806874, 4.04920562286277137153920910164, 4.88365124353271090771579103576, 6.44094173925662194992837631202, 6.75419593028593002724795195065, 8.107454686399614822643316366378, 8.480044338069604170768183440053, 10.10601504538735358614367680089, 10.45872320305995966690063357141

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

\(L(3)\)	\(\approx\)	\(1.687995545\)
\(L(\frac12)\)	\(\approx\)	\(1.687995545\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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