LMFDB - L-function 2-350-5.4-c5-0-36

Dirichlet series

L(s) = 1

+ 4i·2^-s − 3i·3^-s − 16·4^-s + 12·6^-s − 49i·7^-s − 64i·8^-s + 234·9^-s + 405·11^-s + 48i·12^-s − 391i·13^-s + 196·14^-s + 256·16^-s − 999i·17^-s + 936i·18^-s − 2.34e3·19^-s + ⋯

L(s) = 1

+ 0.707i·2^-s − 0.192i·3^-s − 0.5·4^-s + 0.136·6^-s − 0.377i·7^-s − 0.353i·8^-s + 0.962·9^-s + 1.00·11^-s + 0.0962i·12^-s − 0.641i·13^-s + 0.267·14^-s + 0.250·16^-s − 0.838i·17^-s + 0.680i·18^-s − 1.48·19^-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

$L(3)$	$\approx$	$1.551456411$
$L(\frac12)$	$\approx$	$1.551456411$
$L(\frac{7}{2})$		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 + 3iT - 243T^{2} $
	11	$ 1 - 405T + 1.61e5T^{2} $
	13	$ 1 + 391iT - 3.71e5T^{2} $
	17	$ 1 + 999iT - 1.41e6T^{2} $
	19	$ 1 + 2.34e3T + 2.47e6T^{2} $
	23	$ 1 - 2.43e3iT - 6.43e6T^{2} $
	29	$ 1 + 8.25e3T + 2.05e7T^{2} $
	31	$ 1 - 4.01e3T + 2.86e7T^{2} $
	37	$ 1 - 7.04e3iT - 6.93e7T^{2} $
	41	$ 1 - 3.33e3T + 1.15e8T^{2} $
	43	$ 1 + 2.35e4iT - 1.47e8T^{2} $
	47	$ 1 + 1.03e4iT - 2.29e8T^{2} $
	53	$ 1 - 3.08e3iT - 4.18e8T^{2} $
	59	$ 1 - 1.88e4T + 7.14e8T^{2} $
	61	$ 1 - 2.16e4T + 8.44e8T^{2} $
	67	$ 1 + 5.21e4iT - 1.35e9T^{2} $
	71	$ 1 + 2.85e4T + 1.80e9T^{2} $
	73	$ 1 + 7.03e4iT - 2.07e9T^{2} $
	79	$ 1 + 5.88e4T + 3.07e9T^{2} $
	83	$ 1 - 756iT - 3.93e9T^{2} $
	89	$ 1 + 1.35e5T + 5.58e9T^{2} $
	97	$ 1 + 1.10e5iT - 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

−10.30746709885766710239771793117, −9.500413462088595009244174372353, −8.530502277944782146789754948484, −7.41624060770527472970787346996, −6.84495175321044538955280963000, −5.75913212469939274370912961172, −4.52223876740094904572226963669, −3.61827897192229791230810929295, −1.74760237765255382078927926585, −0.41671397945096206055179035081, 1.29871180177463202266503025227, 2.28889899083210710493360341396, 3.93556482073802739511391746779, 4.41217257539128596325672669537, 5.97913053468422386248293348860, 6.92158222362119014219056018762, 8.307446878714609250873939286612, 9.147879500394021129993785859046, 9.924204737832344596010864951707, 10.87524964580864360048042156699

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.551456411\)
\(L(\frac12)\)	\(\approx\)	\(1.551456411\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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