LMFDB - L-function 2-350-5.4-c9-0-77

Dirichlet series

L(s) = 1

− 16i·2^-s − 233. i·3^-s − 256·4^-s − 3.72e3·6^-s + 2.40e3i·7^-s + 4.09e3i·8^-s − 3.46e4·9^-s + 7.28e4·11^-s + 5.96e4i·12^-s − 3.93e4i·13^-s + 3.84e4·14^-s + 6.55e4·16^-s − 5.11e5i·17^-s + 5.54e5i·18^-s − 9.00e5·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 1.66i·3^-s − 0.5·4^-s − 1.17·6^-s + 0.377i·7^-s + 0.353i·8^-s − 1.75·9^-s + 1.50·11^-s + 0.830i·12^-s − 0.382i·13^-s + 0.267·14^-s + 0.250·16^-s − 1.48i·17^-s + 1.24i·18^-s − 1.58·19^-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

$L(5)$	$\approx$	$1.013564079$
$L(\frac12)$	$\approx$	$1.013564079$
$L(\frac{11}{2})$		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 + 233. iT - 1.96e4T^{2} $
	11	$ 1 - 7.28e4T + 2.35e9T^{2} $
	13	$ 1 + 3.93e4iT - 1.06e10T^{2} $
	17	$ 1 + 5.11e5iT - 1.18e11T^{2} $
	19	$ 1 + 9.00e5T + 3.22e11T^{2} $
	23	$ 1 + 3.82e5iT - 1.80e12T^{2} $
	29	$ 1 - 4.73e6T + 1.45e13T^{2} $
	31	$ 1 + 7.93e5T + 2.64e13T^{2} $
	37	$ 1 + 1.72e7iT - 1.29e14T^{2} $
	41	$ 1 + 1.53e7T + 3.27e14T^{2} $
	43	$ 1 + 1.87e7iT - 5.02e14T^{2} $
	47	$ 1 + 4.68e7iT - 1.11e15T^{2} $
	53	$ 1 + 2.06e7iT - 3.29e15T^{2} $
	59	$ 1 - 1.30e7T + 8.66e15T^{2} $
	61	$ 1 - 1.55e8T + 1.16e16T^{2} $
	67	$ 1 - 2.45e8iT - 2.72e16T^{2} $
	71	$ 1 + 3.87e8T + 4.58e16T^{2} $
	73	$ 1 - 3.12e8iT - 5.88e16T^{2} $
	79	$ 1 + 2.41e8T + 1.19e17T^{2} $
	83	$ 1 + 2.00e8iT - 1.86e17T^{2} $
	89	$ 1 + 7.00e7T + 3.50e17T^{2} $
	97	$ 1 - 6.69e8iT - 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

−8.915755678843719384369794696716, −8.454548413348791511599124078897, −7.13535473122084667574169046731, −6.55227140586185896433338949615, −5.44300891514980589434134075907, −4.03941821347905750733225819837, −2.69731485354941333305041959844, −1.97408220018198100474695639194, −0.969740471054021611272947439222, −0.21223485948662850382134065968, 1.45427131476559294672927783850, 3.31316856883532963489562169156, 4.23194130586719902655162960776, 4.58412637070452396629923879165, 6.04951728038595250034195943054, 6.63333524032975339826639296078, 8.269278581549291701207391447733, 8.849147739875570874401589889575, 9.754888394808731715415772064692, 10.45199991832672055157838563056

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.013564079\)
\(L(\frac12)\)	\(\approx\)	\(1.013564079\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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