LMFDB - L-function 2-30e2-3.2-c2-0-7

Dirichlet series

L(s) = 1

+ 7^-s − 4.24i·11^-s + 7·13^-s − 4.24i·17^-s − 7·19^-s − 29.6i·23^-s + 29.6i·29^-s + 17·31^-s + 16·37^-s − 50.9i·41^-s + 55·43^-s − 46.6i·47^-s − 48·49^-s − 84.8i·53^-s + 55.1i·59^-s + ⋯

L(s) = 1

+ 0.142·7^-s − 0.385i·11^-s + 0.538·13^-s − 0.249i·17^-s − 0.368·19^-s − 1.29i·23^-s + 1.02i·29^-s + 0.548·31^-s + 0.432·37^-s − 1.24i·41^-s + 1.27·43^-s − 0.992i·47^-s − 0.979·49^-s − 1.60i·53^-s + 0.934i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$24.5232$
Root analytic conductor:	$4.95209$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (701, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :1),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.763204608$
$L(\frac12)$	$\approx$	$1.763204608$
$L(2)$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 - T + 49T^{2} $
	11	$ 1 + 4.24iT - 121T^{2} $
	13	$ 1 - 7T + 169T^{2} $
	17	$ 1 + 4.24iT - 289T^{2} $
	19	$ 1 + 7T + 361T^{2} $
	23	$ 1 + 29.6iT - 529T^{2} $
	29	$ 1 - 29.6iT - 841T^{2} $
	31	$ 1 - 17T + 961T^{2} $
	37	$ 1 - 16T + 1.36e3T^{2} $
	41	$ 1 + 50.9iT - 1.68e3T^{2} $
	43	$ 1 - 55T + 1.84e3T^{2} $
	47	$ 1 + 46.6iT - 2.20e3T^{2} $
	53	$ 1 + 84.8iT - 2.80e3T^{2} $
	59	$ 1 - 55.1iT - 3.48e3T^{2} $
	61	$ 1 - 65T + 3.72e3T^{2} $
	67	$ 1 - 49T + 4.48e3T^{2} $
	71	$ 1 + 50.9iT - 5.04e3T^{2} $
	73	$ 1 - 88T + 5.32e3T^{2} $
	79	$ 1 + 40T + 6.24e3T^{2} $
	83	$ 1 + 156. iT - 6.88e3T^{2} $
	89	$ 1 - 101. iT - 7.92e3T^{2} $
	97	$ 1 + 41T + 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

−9.833376969806707830951283907990, −8.766715718605769084927511130570, −8.314522074295510227305328740979, −7.16023420339230932141109147780, −6.37186327173828312955884966468, −5.41254394100573927992823592097, −4.40647406764850872835359828941, −3.36124969219007441285993405610, −2.15300780310458775800436869368, −0.65379394620780875393636780049, 1.18927165746010888507052262645, 2.49226740001684189182400988810, 3.75487489311951140460769370037, 4.65297367975343320918081502541, 5.77368050017941388068193082583, 6.53081296600648337006074799463, 7.64449076527837287694459691226, 8.241304989791121879502021766723, 9.340488369545554888684398464750, 9.907467649832000497912187578882

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.763204608\)
\(L(\frac12)\)	\(\approx\)	\(1.763204608\)
\(L(2)\)		not available
\(L(1)\)		not available

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