LMFDB - L-function 2-2175-5.4-c1-0-41

Dirichlet series

L(s) = 1

− 2.56i·2^-s + i·3^-s − 4.56·4^-s + 2.56·6^-s + 3.12i·7^-s + 6.56i·8^-s − 9^-s − 5.56·11^-s − 4.56i·12^-s − 2i·13^-s + 8·14^-s + 7.68·16^-s − 1.12i·17^-s + 2.56i·18^-s + 3.12·19^-s + ⋯

L(s) = 1

− 1.81i·2^-s + 0.577i·3^-s − 2.28·4^-s + 1.04·6^-s + 1.18i·7^-s + 2.31i·8^-s − 0.333·9^-s − 1.67·11^-s − 1.31i·12^-s − 0.554i·13^-s + 2.13·14^-s + 1.92·16^-s − 0.272i·17^-s + 0.603i·18^-s + 0.716·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2175$ = $3 \cdot 5^{2} \cdot 29$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$17.3674$
Root analytic conductor:	$4.16742$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2175} (349, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2175,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.9179690014$
$L(\frac12)$	$\approx$	$0.9179690014$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - iT $
	5	$ 1 $
	29	$ 1 + T $
good	2	$ 1 + 2.56iT - 2T^{2} $
	7	$ 1 - 3.12iT - 7T^{2} $
	11	$ 1 + 5.56T + 11T^{2} $
	13	$ 1 + 2iT - 13T^{2} $
	17	$ 1 + 1.12iT - 17T^{2} $
	19	$ 1 - 3.12T + 19T^{2} $
	23	$ 1 - 2.43iT - 23T^{2} $
	31	$ 1 - 4T + 31T^{2} $
	37	$ 1 + 10.6iT - 37T^{2} $
	41	$ 1 - 10.6T + 41T^{2} $
	43	$ 1 + 4.68iT - 43T^{2} $
	47	$ 1 - 4.87iT - 47T^{2} $
	53	$ 1 + 7.56iT - 53T^{2} $
	59	$ 1 + 12T + 59T^{2} $
	61	$ 1 - 9.12T + 61T^{2} $
	67	$ 1 + 13.3iT - 67T^{2} $
	71	$ 1 + 11.1T + 71T^{2} $
	73	$ 1 + 10.6iT - 73T^{2} $
	79	$ 1 - 12T + 79T^{2} $
	83	$ 1 - 1.56iT - 83T^{2} $
	89	$ 1 - 4.24T + 89T^{2} $
	97	$ 1 + 6.68iT - 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

−9.288147464462274579651194142907, −8.272043484307526779558785349190, −7.61895413525123564445502202020, −5.78055829869145500749852374058, −5.31353517840059737139040302142, −4.55792882557521425906612668622, −3.39826119013099505134550932173, −2.76312201184365734867181228665, −2.09133606814422816123499123628, −0.41124233683428177989139806321, 0.938075068546832454543000935030, 2.78158736611961523639792596892, 4.13209492771647556333732171127, 4.83892974456055030360231832003, 5.65421265092610980269431655960, 6.46517297776241480823529265099, 7.11941902189559581219967915467, 7.77844300012635102264914450806, 8.099707705731057820353778767370, 9.046853585728283829089775159895

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.9179690014\)
\(L(\frac12)\)	\(\approx\)	\(0.9179690014\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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