LMFDB - L-function 2-2268-21.20-c1-0-25

Dirichlet series

L(s) = 1

− 0.553·5^-s + (2.50 + 0.847i)7^-s − 4.65i·11^-s + 4.13i·13^-s − 7.24·17^-s − 6.71i·19^-s − 5.60i·23^-s − 4.69·25^-s − 1.34i·29^-s − 0.959i·31^-s + (−1.38 − 0.469i)35^-s − 7.06·37^-s + 4.78·41^-s + 2.05·43^-s + 9.80·47^-s + ⋯

L(s) = 1

− 0.247·5^-s + (0.947 + 0.320i)7^-s − 1.40i·11^-s + 1.14i·13^-s − 1.75·17^-s − 1.54i·19^-s − 1.16i·23^-s − 0.938·25^-s − 0.250i·29^-s − 0.172i·31^-s + (−0.234 − 0.0793i)35^-s − 1.16·37^-s + 0.746·41^-s + 0.313·43^-s + 1.43·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2268$ = $2^{2} \cdot 3^{4} \cdot 7$
Sign:	$-0.320 + 0.947i$
Analytic conductor:	$18.1100$
Root analytic conductor:	$4.25559$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2268} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2268,\ (\ :1/2),\ -0.320 + 0.947i)$

Particular Values

$L(1)$	$\approx$	$1.117818987$
$L(\frac12)$	$\approx$	$1.117818987$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-2.50 - 0.847i)T $
good	5	$ 1 + 0.553T + 5T^{2} $
	11	$ 1 + 4.65iT - 11T^{2} $
	13	$ 1 - 4.13iT - 13T^{2} $
	17	$ 1 + 7.24T + 17T^{2} $
	19	$ 1 + 6.71iT - 19T^{2} $
	23	$ 1 + 5.60iT - 23T^{2} $
	29	$ 1 + 1.34iT - 29T^{2} $
	31	$ 1 + 0.959iT - 31T^{2} $
	37	$ 1 + 7.06T + 37T^{2} $
	41	$ 1 - 4.78T + 41T^{2} $
	43	$ 1 - 2.05T + 43T^{2} $
	47	$ 1 - 9.80T + 47T^{2} $
	53	$ 1 + 8.43iT - 53T^{2} $
	59	$ 1 + 7.79T + 59T^{2} $
	61	$ 1 - 6.20iT - 61T^{2} $
	67	$ 1 + 3.37T + 67T^{2} $
	71	$ 1 + 0.407iT - 71T^{2} $
	73	$ 1 + 8.63iT - 73T^{2} $
	79	$ 1 - 0.636T + 79T^{2} $
	83	$ 1 + 5.57T + 83T^{2} $
	89	$ 1 + 6.93T + 89T^{2} $
	97	$ 1 + 8.64iT - 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

−8.908788599768631178611674228821, −8.205683798034699848772431303344, −7.18809182643720012774509218527, −6.49522387502335536399689688166, −5.65962671187768548996925284199, −4.59179554519527590773475061483, −4.15663388332712436127672921632, −2.76226873875508987912490740820, −1.93887767921319368500389726203, −0.37262007912241584520852291056, 1.46310950694804260062959378342, 2.31518455600711260726211999395, 3.73141549516778962068165483910, 4.38932695646819249493350252080, 5.23915703403832122783533394987, 6.04997644313708528174598080728, 7.27045092602670891035322651640, 7.58509754556767378411877035166, 8.376018565467650762171333380560, 9.230774909413522463874875365551

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.117818987\)
\(L(\frac12)\)	\(\approx\)	\(1.117818987\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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