LMFDB - L-function 2-950-95.94-c2-0-44

Dirichlet series

L(s) = 1

− 1.41·2^-s − 1.82·3^-s + 2.00·4^-s + 2.57·6^-s − 13.1i·7^-s − 2.82·8^-s − 5.67·9^-s + 9.79·11^-s − 3.64·12^-s + 10.8·13^-s + 18.5i·14^-s + 4.00·16^-s − 13.1i·17^-s + 8.02·18^-s + (7.67 − 17.3i)19^-s + ⋯

L(s) = 1

− 0.707·2^-s − 0.607·3^-s + 0.500·4^-s + 0.429·6^-s − 1.87i·7^-s − 0.353·8^-s − 0.630·9^-s + 0.890·11^-s − 0.303·12^-s + 0.835·13^-s + 1.32i·14^-s + 0.250·16^-s − 0.770i·17^-s + 0.445·18^-s + (0.403 − 0.914i)19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$950$ = $2 \cdot 5^{2} \cdot 19$
Sign:	$-0.770 + 0.637i$
Analytic conductor:	$25.8856$
Root analytic conductor:	$5.08779$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{950} (949, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 950,\ (\ :1),\ -0.770 + 0.637i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8332866021$
$L(\frac12)$	$\approx$	$0.8332866021$
$L(2)$		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 + 1.41T $
	5	$ 1 $
	19	$ 1 + (-7.67 + 17.3i)T $
good	3	$ 1 + 1.82T + 9T^{2} $
	7	$ 1 + 13.1iT - 49T^{2} $
	11	$ 1 - 9.79T + 121T^{2} $
	13	$ 1 - 10.8T + 169T^{2} $
	17	$ 1 + 13.1iT - 289T^{2} $
	23	$ 1 + 18.1iT - 529T^{2} $
	29	$ 1 - 44.8iT - 841T^{2} $
	31	$ 1 + 11.5iT - 961T^{2} $
	37	$ 1 - 26.1T + 1.36e3T^{2} $
	41	$ 1 - 13.5iT - 1.68e3T^{2} $
	43	$ 1 + 33.0iT - 1.84e3T^{2} $
	47	$ 1 - 7.65iT - 2.20e3T^{2} $
	53	$ 1 + 51.5T + 2.80e3T^{2} $
	59	$ 1 - 28.6iT - 3.48e3T^{2} $
	61	$ 1 - 100.T + 3.72e3T^{2} $
	67	$ 1 - 11.7T + 4.48e3T^{2} $
	71	$ 1 + 126. iT - 5.04e3T^{2} $
	73	$ 1 + 62.4iT - 5.32e3T^{2} $
	79	$ 1 - 45.2iT - 6.24e3T^{2} $
	83	$ 1 - 32.4iT - 6.88e3T^{2} $
	89	$ 1 - 19.4iT - 7.92e3T^{2} $
	97	$ 1 + 166.T + 9.40e3T^{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

−9.533382966299097247385427323818, −8.771439942834387099436841649008, −7.79140861740393021452065632015, −6.85895034826958477948018413774, −6.48448955712422316649249831516, −5.16030660717630990373527877892, −4.10959815882177276312392126927, −3.05890409504515317771923640972, −1.22002956177148842707245501198, −0.43426908167158389191870094187, 1.37268588941395413580630526815, 2.54473405259346576507297586090, 3.74797685039919921968075080580, 5.36354465949569337944189094999, 5.99183505063537353332262969788, 6.45762396535153129976625457496, 8.009716117713368954021526688171, 8.527855907587235916208870171063, 9.281779992808236387411981426847, 9.994817407425720772564590037553

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8332866021\)
\(L(\frac12)\)	\(\approx\)	\(0.8332866021\)
\(L(2)\)		not available
\(L(1)\)		not available

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