LMFDB - L-function 2-675-5.4-c3-0-22

Dirichlet series

L(s) = 1

− 4.72i·2^-s − 14.3·4^-s − 17.6i·7^-s + 29.9i·8^-s + 34.2·11^-s + 53.8i·13^-s − 83.3·14^-s + 26.9·16^-s + 74.7i·17^-s + 89.5·19^-s − 161. i·22^-s + 176. i·23^-s + 254.·26^-s + 252. i·28^-s − 194.·29^-s + ⋯

L(s) = 1

− 1.67i·2^-s − 1.79·4^-s − 0.951i·7^-s + 1.32i·8^-s + 0.938·11^-s + 1.14i·13^-s − 1.59·14^-s + 0.420·16^-s + 1.06i·17^-s + 1.08·19^-s − 1.56i·22^-s + 1.59i·23^-s + 1.92·26^-s + 1.70i·28^-s − 1.24·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.615501544$
$L(\frac12)$	$\approx$	$1.615501544$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	2	$ 1 + 4.72iT - 8T^{2} $
	7	$ 1 + 17.6iT - 343T^{2} $
	11	$ 1 - 34.2T + 1.33e3T^{2} $
	13	$ 1 - 53.8iT - 2.19e3T^{2} $
	17	$ 1 - 74.7iT - 4.91e3T^{2} $
	19	$ 1 - 89.5T + 6.85e3T^{2} $
	23	$ 1 - 176. iT - 1.21e4T^{2} $
	29	$ 1 + 194.T + 2.43e4T^{2} $
	31	$ 1 - 107.T + 2.97e4T^{2} $
	37	$ 1 - 430. iT - 5.06e4T^{2} $
	41	$ 1 + 108.T + 6.89e4T^{2} $
	43	$ 1 - 409. iT - 7.95e4T^{2} $
	47	$ 1 + 409. iT - 1.03e5T^{2} $
	53	$ 1 + 24.7iT - 1.48e5T^{2} $
	59	$ 1 - 295.T + 2.05e5T^{2} $
	61	$ 1 - 305.T + 2.26e5T^{2} $
	67	$ 1 + 915. iT - 3.00e5T^{2} $
	71	$ 1 - 228.T + 3.57e5T^{2} $
	73	$ 1 + 158. iT - 3.89e5T^{2} $
	79	$ 1 - 319.T + 4.93e5T^{2} $
	83	$ 1 + 936. iT - 5.71e5T^{2} $
	89	$ 1 + 920.T + 7.04e5T^{2} $
	97	$ 1 - 914. iT - 9.12e5T^{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

−9.890246438914951135119908481204, −9.596964696105042247489540823285, −8.583154429757997477454711383541, −7.39103635447807507299054738688, −6.39243419269281888849237557623, −4.90638325959927265424513382397, −3.90185685660007436088474209606, −3.41636113982084324442997689643, −1.78325951662628417579985383679, −1.12756400011564122002580853296, 0.53926782065493655482870811141, 2.60562890121891059890999962890, 4.06319475244828915353693545932, 5.32045787217768819356582321616, 5.69187978187702725561380286906, 6.78103668285957476586936976209, 7.46943339354265445160166596047, 8.433185225740946193135386596185, 9.059826275274318181205831242147, 9.793777218525966946865272842090

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

\(L(2)\)	\(\approx\)	\(1.615501544\)
\(L(\frac12)\)	\(\approx\)	\(1.615501544\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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