LMFDB - L-function 2-3330-5.4-c1-0-22

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s + (−2.07 + 0.826i)5^-s + 4.67i·7^-s + i·8^-s + (0.826 + 2.07i)10^-s − 0.0451·11^-s − 5.26i·13^-s + 4.67·14^-s + 16^-s − 3.60i·17^-s + 6.22·19^-s + (2.07 − 0.826i)20^-s + 0.0451i·22^-s − 2.20i·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + (−0.929 + 0.369i)5^-s + 1.76i·7^-s + 0.353i·8^-s + (0.261 + 0.656i)10^-s − 0.0136·11^-s − 1.46i·13^-s + 1.24·14^-s + 0.250·16^-s − 0.875i·17^-s + 1.42·19^-s + (0.464 − 0.184i)20^-s + 0.00962i·22^-s − 0.459i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3330$ = $2 \cdot 3^{2} \cdot 5 \cdot 37$
Sign:	$0.929 - 0.369i$
Analytic conductor:	$26.5901$
Root analytic conductor:	$5.15656$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3330} (1999, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3330,\ (\ :1/2),\ 0.929 - 0.369i)$

Particular Values

$L(1)$	$\approx$	$1.242385349$
$L(\frac12)$	$\approx$	$1.242385349$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	5	$ 1 + (2.07 - 0.826i)T $
	37	$ 1 - iT $
good	7	$ 1 - 4.67iT - 7T^{2} $
	11	$ 1 + 0.0451T + 11T^{2} $
	13	$ 1 + 5.26iT - 13T^{2} $
	17	$ 1 + 3.60iT - 17T^{2} $
	19	$ 1 - 6.22T + 19T^{2} $
	23	$ 1 + 2.20iT - 23T^{2} $
	29	$ 1 + 4.20T + 29T^{2} $
	31	$ 1 + 3.01T + 31T^{2} $
	41	$ 1 - 7.38T + 41T^{2} $
	43	$ 1 - 5.54iT - 43T^{2} $
	47	$ 1 - 4.28iT - 47T^{2} $
	53	$ 1 - 6.10iT - 53T^{2} $
	59	$ 1 - 13.5T + 59T^{2} $
	61	$ 1 + 4.27T + 61T^{2} $
	67	$ 1 - 12.3iT - 67T^{2} $
	71	$ 1 - 4.49T + 71T^{2} $
	73	$ 1 + 7.03iT - 73T^{2} $
	79	$ 1 - 8.72T + 79T^{2} $
	83	$ 1 + 0.880iT - 83T^{2} $
	89	$ 1 - 9.97T + 89T^{2} $
	97	$ 1 - 0.240iT - 97T^{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

−8.841400960309479982937635207151, −7.87742708157636472854583974888, −7.55078728251489641186206074181, −6.26494207915933409499450137949, −5.43043911396622875897828813254, −4.95614572021263443641393898013, −3.70414797217783657629331417126, −2.91429992809403701656967970517, −2.47783655816558234561235745211, −0.859819334857564487901276513036, 0.53940026411688828613713336718, 1.62344997431544734599408318251, 3.63398472270167328905437470923, 3.84865184716299926590080838945, 4.65803506329269988787531628205, 5.49789391307077471892364960446, 6.64988435139270175694502076849, 7.22041709880022152244581660733, 7.61794498378735648076081707490, 8.379327814305137717946749053790

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.242385349\)
\(L(\frac12)\)	\(\approx\)	\(1.242385349\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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