LMFDB - L-function 2-7524-209.208-c1-0-9

Dirichlet series

L(s) = 1

− 3.31i·11^-s − 7.15·13^-s − 8.19i·17^-s − 4.35·19^-s − 5·25^-s + 7·49^-s + 5.72i·53^-s + 11.7i·59^-s − 0.271i·71^-s + 12.7·79^-s + 14.8i·83^-s + 17.7i·89^-s + 13.2i·101^-s + 17.4·109^-s − 6.27i·113^-s + ⋯

L(s) = 1

− 1.00i·11^-s − 1.98·13^-s − 1.98i·17^-s − 1.00·19^-s − 25^-s + 49^-s + 0.786i·53^-s + 1.52i·59^-s − 0.0322i·71^-s + 1.43·79^-s + 1.62i·83^-s + 1.87i·89^-s + 1.32i·101^-s + 1.67·109^-s − 0.589i·113^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7524$ = $2^{2} \cdot 3^{2} \cdot 11 \cdot 19$
Sign:	$-i$
Analytic conductor:	$60.0794$
Root analytic conductor:	$7.75109$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7524} (2089, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7524,\ (\ :1/2),\ -i)$

Particular Values

$L(1)$	$\approx$	$0.5074676483$
$L(\frac12)$	$\approx$	$0.5074676483$
$L(\frac{3}{2})$		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 $
	11	$ 1 + 3.31iT $
	19	$ 1 + 4.35T $
good	5	$ 1 + 5T^{2} $
	7	$ 1 - 7T^{2} $
	13	$ 1 + 7.15T + 13T^{2} $
	17	$ 1 + 8.19iT - 17T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 5.72iT - 53T^{2} $
	59	$ 1 - 11.7iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 + 0.271iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 12.7T + 79T^{2} $
	83	$ 1 - 14.8iT - 83T^{2} $
	89	$ 1 - 17.7iT - 89T^{2} $
	97	$ 1 - 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

−7.916398398972334653878316954348, −7.38405145193390019456276758844, −6.79873181681668687978814341652, −5.90822971973235372907942822747, −5.20514749607580846545961337502, −4.62725052403952637073409562087, −3.75038006020192462164685632467, −2.67688571323049050710653110020, −2.33795129951428205460981606912, −0.793693308409649463217673520114, 0.14694924778403813915291489496, 1.93017960363261647183386561467, 2.12940650971654575009921013153, 3.39657232461308309903623292875, 4.28015134027379989357709853376, 4.73454924857961062174490970101, 5.64394553341733118307269293040, 6.35467065869593966519331845934, 7.08319194477117176343317431110, 7.69134769375573005604325875534

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5074676483\)
\(L(\frac12)\)	\(\approx\)	\(0.5074676483\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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