LMFDB - L-function 2-3344-76.75-c1-0-2

Dirichlet series

L(s) = 1

− 3.25·3^-s − 2.27·5^-s + 4.28i·7^-s + 7.58·9^-s − i·11^-s − 2.82i·13^-s + 7.39·15^-s − 5.42·17^-s + (−4.27 − 0.826i)19^-s − 13.9i·21^-s + 8.87i·23^-s + 0.169·25^-s − 14.9·27^-s + 5.91i·29^-s + 5.48·31^-s + ⋯

L(s) = 1

− 1.87·3^-s − 1.01·5^-s + 1.61i·7^-s + 2.52·9^-s − 0.301i·11^-s − 0.783i·13^-s + 1.90·15^-s − 1.31·17^-s + (−0.981 − 0.189i)19^-s − 3.03i·21^-s + 1.85i·23^-s + 0.0338·25^-s − 2.86·27^-s + 1.09i·29^-s + 0.985·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3344$ = $2^{4} \cdot 11 \cdot 19$
Sign:	$-0.189 + 0.981i$
Analytic conductor:	$26.7019$
Root analytic conductor:	$5.16739$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3344} (1519, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3344,\ (\ :1/2),\ -0.189 + 0.981i)$

Particular Values

$L(1)$	$\approx$	$0.06910576641$
$L(\frac12)$	$\approx$	$0.06910576641$
$L(\frac{3}{2})$		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 $
	11	$ 1 + iT $
	19	$ 1 + (4.27 + 0.826i)T $
good	3	$ 1 + 3.25T + 3T^{2} $
	5	$ 1 + 2.27T + 5T^{2} $
	7	$ 1 - 4.28iT - 7T^{2} $
	13	$ 1 + 2.82iT - 13T^{2} $
	17	$ 1 + 5.42T + 17T^{2} $
	23	$ 1 - 8.87iT - 23T^{2} $
	29	$ 1 - 5.91iT - 29T^{2} $
	31	$ 1 - 5.48T + 31T^{2} $
	37	$ 1 - 3.25iT - 37T^{2} $
	41	$ 1 + 2.92iT - 41T^{2} $
	43	$ 1 - 9.28iT - 43T^{2} $
	47	$ 1 - 5.09iT - 47T^{2} $
	53	$ 1 - 3.16iT - 53T^{2} $
	59	$ 1 - 1.30T + 59T^{2} $
	61	$ 1 - 1.77T + 61T^{2} $
	67	$ 1 + 8.89T + 67T^{2} $
	71	$ 1 - 11.2T + 71T^{2} $
	73	$ 1 + 5.70T + 73T^{2} $
	79	$ 1 + 15.6T + 79T^{2} $
	83	$ 1 - 14.9iT - 83T^{2} $
	89	$ 1 - 18.0iT - 89T^{2} $
	97	$ 1 + 14.6iT - 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

−9.150806533998677104858013790938, −8.361953169312776899363841145891, −7.58004691587833534625501896662, −6.66503965421320249064119583638, −6.08352760427470221738244714807, −5.42160963779675762610860376786, −4.80405832790401849036278361428, −3.97262182539143848029558989592, −2.76475226704350873505233957454, −1.40958688616565043274417771073, 0.05158145100042320016333143287, 0.60173286025156759294585305971, 2.00394841078946402544645504085, 3.94946183339123993067478924585, 4.39596443374471378359212160268, 4.62438419633564003138349876480, 6.01761391472218047522401925720, 6.75899717701067406147820964807, 6.96008093019622096269412387559, 7.83862033362544639038001549707

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.06910576641\)
\(L(\frac12)\)	\(\approx\)	\(0.06910576641\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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