LMFDB - L-function 2-2175-87.86-c0-0-4

Dirichlet series

L(s) = 1

− 0.347·2^-s − 3^-s − 0.879·4^-s + 0.347·6^-s + 1.87·7^-s + 0.652·8^-s + 9^-s + 1.53·11^-s + 0.879·12^-s − 0.347·13^-s − 0.652·14^-s + 0.652·16^-s − 1.53·17^-s − 0.347·18^-s − 1.87·21^-s − 0.532·22^-s − 0.652·24^-s + 0.120·26^-s − 27^-s − 1.65·28^-s + 29^-s − 0.879·32^-s − 1.53·33^-s + 0.532·34^-s − 0.879·36^-s + 0.347·39^-s − 41^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2175$ = $3 \cdot 5^{2} \cdot 29$
Sign:	$1$
Analytic conductor:	$1.08546$
Root analytic conductor:	$1.04185$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2175} (1826, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(2,\ 2175,\ (\ :0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7772536054$
$L(\frac12)$	$\approx$	$0.7772536054$
$L(1)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 + T $
	5	$ 1 $
	29	$ 1 - T $
good	2	$ 1 + 0.347T + T^{2} $
	7	$ 1 - 1.87T + T^{2} $
	11	$ 1 - 1.53T + T^{2} $
	13	$ 1 + 0.347T + T^{2} $
	17	$ 1 + 1.53T + T^{2} $
	19	$ 1 - T^{2} $
	23	$ 1 - T^{2} $
	31	$ 1 - T^{2} $
	37	$ 1 - T^{2} $
	41	$ 1 + T + T^{2} $
	43	$ 1 - T^{2} $
	47	$ 1 - 1.87T + T^{2} $
	53	$ 1 - T^{2} $
	59	$ 1 - T^{2} $
	61	$ 1 - T^{2} $
	67	$ 1 + 1.53T + T^{2} $
	71	$ 1 - T^{2} $
	73	$ 1 - T^{2} $
	79	$ 1 - T^{2} $
	83	$ 1 - T^{2} $
	89	$ 1 - 0.347T + T^{2} $
	97	$ 1 - T^{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

−8.991312518695984607772139445866, −8.739276093829901139504284956686, −7.71621506664115351524724123027, −6.99658498534019354268492847128, −6.06636784162407590066444391644, −5.05295758961349030166920206413, −4.53080275361845858840565894443, −4.02100546134468548233210777415, −1.93003054497361431391125367017, −1.05380231752988230136180855778, 1.05380231752988230136180855778, 1.93003054497361431391125367017, 4.02100546134468548233210777415, 4.53080275361845858840565894443, 5.05295758961349030166920206413, 6.06636784162407590066444391644, 6.99658498534019354268492847128, 7.71621506664115351524724123027, 8.739276093829901139504284956686, 8.991312518695984607772139445866

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7772536054\)
\(L(\frac12)\)	\(\approx\)	\(0.7772536054\)
\(L(1)\)		not available
\(L(1)\)		not available

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