LMFDB - L-function 2-1512-21.20-c1-0-22

Dirichlet series

L(s) = 1

+ 2.85·5^-s + (−2.30 + 1.29i)7^-s − 3.55i·11^-s − 2.43i·13^-s − 0.860·17^-s − 3.55i·19^-s − 5.94i·23^-s + 3.15·25^-s + 2.88i·29^-s − 9.01i·31^-s + (−6.58 + 3.70i)35^-s + 7.43·37^-s + 6.85·41^-s + 3.48·43^-s − 0.263·47^-s + ⋯

L(s) = 1

+ 1.27·5^-s + (−0.871 + 0.489i)7^-s − 1.07i·11^-s − 0.675i·13^-s − 0.208·17^-s − 0.816i·19^-s − 1.23i·23^-s + 0.631·25^-s + 0.535i·29^-s − 1.61i·31^-s + (−1.11 + 0.625i)35^-s + 1.22·37^-s + 1.07·41^-s + 0.532·43^-s − 0.0384·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1512$ = $2^{3} \cdot 3^{3} \cdot 7$
Sign:	$0.489 + 0.871i$
Analytic conductor:	$12.0733$
Root analytic conductor:	$3.47467$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1512} (377, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1512,\ (\ :1/2),\ 0.489 + 0.871i)$

Particular Values

$L(1)$	$\approx$	$1.756942169$
$L(\frac12)$	$\approx$	$1.756942169$
$L(\frac{3}{2})$		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 + (2.30 - 1.29i)T $
good	5	$ 1 - 2.85T + 5T^{2} $
	11	$ 1 + 3.55iT - 11T^{2} $
	13	$ 1 + 2.43iT - 13T^{2} $
	17	$ 1 + 0.860T + 17T^{2} $
	19	$ 1 + 3.55iT - 19T^{2} $
	23	$ 1 + 5.94iT - 23T^{2} $
	29	$ 1 - 2.88iT - 29T^{2} $
	31	$ 1 + 9.01iT - 31T^{2} $
	37	$ 1 - 7.43T + 37T^{2} $
	41	$ 1 - 6.85T + 41T^{2} $
	43	$ 1 - 3.48T + 43T^{2} $
	47	$ 1 + 0.263T + 47T^{2} $
	53	$ 1 - 7.76iT - 53T^{2} $
	59	$ 1 + 9.72T + 59T^{2} $
	61	$ 1 - 1.62iT - 61T^{2} $
	67	$ 1 - 15.0T + 67T^{2} $
	71	$ 1 - 15.1iT - 71T^{2} $
	73	$ 1 + 9.20iT - 73T^{2} $
	79	$ 1 + 7.55T + 79T^{2} $
	83	$ 1 - 0.922T + 83T^{2} $
	89	$ 1 + 4.90T + 89T^{2} $
	97	$ 1 + 10.1iT - 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

−9.316663932921178690942006206601, −8.795124553499220359068196157846, −7.80081124394252236690843979359, −6.61945435612860183739615433680, −5.99436953762969303079588294629, −5.54184457169374307258340623771, −4.28284793172589528170139559393, −2.93024969544967206436352925671, −2.40536111560383700892023433738, −0.70398043192168375943062518070, 1.44642254955160223267636647396, 2.38822533076062067935965779125, 3.61092439083925538511104857707, 4.59692579510502018989812374081, 5.63301602445863761030473059951, 6.37047998736714836543254999436, 7.04065617047545025306628203302, 7.909388078503030264691673262783, 9.242179115138732445924696140941, 9.565826677346086050032204814886

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.756942169\)
\(L(\frac12)\)	\(\approx\)	\(1.756942169\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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