LMFDB - L-function 2-6900-5.4-c1-0-22

Dirichlet series

L(s) = 1

+ i·3^-s + 3.73i·7^-s − 9^-s + 4.84·11^-s − 2.84i·13^-s − 0.890i·17^-s − 6.84·19^-s − 3.73·21^-s + i·23^-s − i·27^-s + 0.890·29^-s + 7.73·31^-s + 4.84i·33^-s + 1.95i·37^-s + 2.84·39^-s + ⋯

L(s) = 1

+ 0.577i·3^-s + 1.41i·7^-s − 0.333·9^-s + 1.46·11^-s − 0.788i·13^-s − 0.216i·17^-s − 1.57·19^-s − 0.815·21^-s + 0.208i·23^-s − 0.192i·27^-s + 0.165·29^-s + 1.38·31^-s + 0.843i·33^-s + 0.321i·37^-s + 0.455·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6900$ = $2^{2} \cdot 3 \cdot 5^{2} \cdot 23$
Sign:	$-0.447 - 0.894i$
Analytic conductor:	$55.0967$
Root analytic conductor:	$7.42272$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6900} (6349, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6900,\ (\ :1/2),\ -0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.893588016$
$L(\frac12)$	$\approx$	$1.893588016$
$L(\frac{3}{2})$		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 $
	3	$ 1 - iT $
	5	$ 1 $
	23	$ 1 - iT $
good	7	$ 1 - 3.73iT - 7T^{2} $
	11	$ 1 - 4.84T + 11T^{2} $
	13	$ 1 + 2.84iT - 13T^{2} $
	17	$ 1 + 0.890iT - 17T^{2} $
	19	$ 1 + 6.84T + 19T^{2} $
	29	$ 1 - 0.890T + 29T^{2} $
	31	$ 1 - 7.73T + 31T^{2} $
	37	$ 1 - 1.95iT - 37T^{2} $
	41	$ 1 - 12.3T + 41T^{2} $
	43	$ 1 + 3.47iT - 43T^{2} $
	47	$ 1 - 6.62iT - 47T^{2} $
	53	$ 1 - 12.3iT - 53T^{2} $
	59	$ 1 + 0.890T + 59T^{2} $
	61	$ 1 - 8.62T + 61T^{2} $
	67	$ 1 + 7.73iT - 67T^{2} $
	71	$ 1 + 12.3T + 71T^{2} $
	73	$ 1 - 16.5iT - 73T^{2} $
	79	$ 1 + 13.4T + 79T^{2} $
	83	$ 1 + 4.58iT - 83T^{2} $
	89	$ 1 - 15.1T + 89T^{2} $
	97	$ 1 - 17.2iT - 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

−8.365719591862374570409166827811, −7.62228191271879195773591637053, −6.45234234593840437137028992978, −6.14489138398912529329745345994, −5.42438870123482264960345241676, −4.51560514663681098563470383850, −3.97890220102551953194422674026, −2.89851172526527792771628969114, −2.34743381211373651102793988624, −1.08893143056791983158340530130, 0.52383143579804121441852865610, 1.40903600137974720641253581772, 2.24155923545212858575972145437, 3.45472471047459043303472096004, 4.29984523927225054758926317660, 4.45773410330598997222157509356, 5.94402195935203254554535772517, 6.50935939506903110793921014456, 6.93219320322448361996715372709, 7.58418274410272815327829534243

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.893588016\)
\(L(\frac12)\)	\(\approx\)	\(1.893588016\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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