LMFDB - L-function 2-4032-168.125-c1-0-13

Dirichlet series

L(s) = 1

− 3.91i·5^-s + (2.03 + 1.68i)7^-s − 1.62·11^-s − 2.54·13^-s + 2.94·17^-s − 2.72·19^-s + 8.57i·23^-s − 10.3·25^-s − 8.21·29^-s + 8.08i·31^-s + (6.61 − 7.98i)35^-s + 8.05i·37^-s + 9.17·41^-s + 9.05i·43^-s + 1.17·47^-s + ⋯

L(s) = 1

− 1.75i·5^-s + (0.769 + 0.638i)7^-s − 0.490·11^-s − 0.706·13^-s + 0.713·17^-s − 0.624·19^-s + 1.78i·23^-s − 2.07·25^-s − 1.52·29^-s + 1.45i·31^-s + (1.11 − 1.34i)35^-s + 1.32i·37^-s + 1.43·41^-s + 1.38i·43^-s + 0.171·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.331 - 0.943i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.331 - 0.943i)$

Particular Values

$L(1)$	$\approx$	$1.145629986$
$L(\frac12)$	$\approx$	$1.145629986$
$L(\frac{3}{2})$		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 + (-2.03 - 1.68i)T $
good	5	$ 1 + 3.91iT - 5T^{2} $
	11	$ 1 + 1.62T + 11T^{2} $
	13	$ 1 + 2.54T + 13T^{2} $
	17	$ 1 - 2.94T + 17T^{2} $
	19	$ 1 + 2.72T + 19T^{2} $
	23	$ 1 - 8.57iT - 23T^{2} $
	29	$ 1 + 8.21T + 29T^{2} $
	31	$ 1 - 8.08iT - 31T^{2} $
	37	$ 1 - 8.05iT - 37T^{2} $
	41	$ 1 - 9.17T + 41T^{2} $
	43	$ 1 - 9.05iT - 43T^{2} $
	47	$ 1 - 1.17T + 47T^{2} $
	53	$ 1 - 2.44T + 53T^{2} $
	59	$ 1 - 1.45iT - 59T^{2} $
	61	$ 1 + 9.74T + 61T^{2} $
	67	$ 1 - 7.35iT - 67T^{2} $
	71	$ 1 + 7.71iT - 71T^{2} $
	73	$ 1 + 15.0iT - 73T^{2} $
	79	$ 1 + 0.0913T + 79T^{2} $
	83	$ 1 + 2.03iT - 83T^{2} $
	89	$ 1 - 1.48T + 89T^{2} $
	97	$ 1 - 7.53iT - 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

−8.632480258796694359652553826846, −7.78187876478445137919203119376, −7.58966483758230542968252682693, −6.08336832398163391567201422582, −5.39116692812402808485406205924, −4.97801832230655773888456385795, −4.28124475874925060172246837835, −3.12475281449632238403291192372, −1.87997134883289180443305919549, −1.22345275234781909430143803482, 0.32660685071541513394655517477, 2.14808102068937957117901224179, 2.55950290419127255546798468423, 3.73921961189236619970195839279, 4.30300716086080619097966312731, 5.45343399486159404339662764243, 6.13486968566783735912103472782, 7.03945840192834231664060619720, 7.49712040570707270290441254082, 7.971871374634968007054397146336

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.145629986\)
\(L(\frac12)\)	\(\approx\)	\(1.145629986\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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