LMFDB - L-function 2-350-5.4-c5-0-18

Dirichlet series

L(s) = 1

− 4i·2^-s − 17i·3^-s − 16·4^-s − 68·6^-s + 49i·7^-s + 64i·8^-s − 46·9^-s − 715·11^-s + 272i·12^-s + 331i·13^-s + 196·14^-s + 256·16^-s + 1.69e3i·17^-s + 184i·18^-s + 1.71e3·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 1.09i·3^-s − 0.5·4^-s − 0.771·6^-s + 0.377i·7^-s + 0.353i·8^-s − 0.189·9^-s − 1.78·11^-s + 0.545i·12^-s + 0.543i·13^-s + 0.267·14^-s + 0.250·16^-s + 1.42i·17^-s + 0.133i·18^-s + 1.09·19^-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

$L(3)$	$\approx$	$1.588386381$
$L(\frac12)$	$\approx$	$1.588386381$
$L(\frac{7}{2})$		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 + 17iT - 243T^{2} $
	11	$ 1 + 715T + 1.61e5T^{2} $
	13	$ 1 - 331iT - 3.71e5T^{2} $
	17	$ 1 - 1.69e3iT - 1.41e6T^{2} $
	19	$ 1 - 1.71e3T + 2.47e6T^{2} $
	23	$ 1 + 3.95e3iT - 6.43e6T^{2} $
	29	$ 1 + 4.57e3T + 2.05e7T^{2} $
	31	$ 1 - 6.75e3T + 2.86e7T^{2} $
	37	$ 1 - 1.65e4iT - 6.93e7T^{2} $
	41	$ 1 - 1.88e4T + 1.15e8T^{2} $
	43	$ 1 - 2.25e3iT - 1.47e8T^{2} $
	47	$ 1 - 537iT - 2.29e8T^{2} $
	53	$ 1 + 1.09e4iT - 4.18e8T^{2} $
	59	$ 1 - 2.59e4T + 7.14e8T^{2} $
	61	$ 1 - 3.91e4T + 8.44e8T^{2} $
	67	$ 1 + 4.41e3iT - 1.35e9T^{2} $
	71	$ 1 + 3.18e4T + 1.80e9T^{2} $
	73	$ 1 + 5.01e3iT - 2.07e9T^{2} $
	79	$ 1 - 2.79e4T + 3.07e9T^{2} $
	83	$ 1 - 3.76e4iT - 3.93e9T^{2} $
	89	$ 1 - 1.72e4T + 5.58e9T^{2} $
	97	$ 1 - 6.31e4iT - 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

−10.54164765101383559841887780534, −9.789823787617255118432247411140, −8.381581213438818775333932740666, −7.903656042881116029370356597644, −6.68627590212716470361602700298, −5.62237699697275029917362230310, −4.43241809883464917781612230318, −2.85038191261791535347148887050, −2.01283574758085709764277292845, −0.800794083188094380774345139492, 0.58102727409788131478990997147, 2.81557602060556826075794594913, 3.94805097309223833837832440191, 5.17148082133071264803394222048, 5.51441757194966960079643938234, 7.39442305784458136179207890557, 7.67772656745840568474321369764, 9.176747166514246907711093275198, 9.799375259608524416022015197766, 10.57752443192637267844597930175

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.588386381\)
\(L(\frac12)\)	\(\approx\)	\(1.588386381\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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