LMFDB - L-function 2-15e2-3.2-c4-0-3

Dirichlet series

L(s) = 1

− 7.84i·2^-s − 45.5·4^-s − 71.8·7^-s + 231. i·8^-s − 66.0i·11^-s − 143.·13^-s + 563. i·14^-s + 1.09e3·16^-s − 88.1i·17^-s + 397.·19^-s − 518.·22^-s + 189. i·23^-s + 1.12e3i·26^-s + 3.27e3·28^-s − 649. i·29^-s + ⋯

L(s) = 1

− 1.96i·2^-s − 2.84·4^-s − 1.46·7^-s + 3.62i·8^-s − 0.545i·11^-s − 0.850·13^-s + 2.87i·14^-s + 4.25·16^-s − 0.304i·17^-s + 1.10·19^-s − 1.07·22^-s + 0.357i·23^-s + 1.66i·26^-s + 4.17·28^-s − 0.772i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$23.2582$
Root analytic conductor:	$4.82267$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (26, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :2),\ 0.577 + 0.816i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.6396675728$
$L(\frac12)$	$\approx$	$0.6396675728$
$L(3)$		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 + 7.84iT - 16T^{2} $
	7	$ 1 + 71.8T + 2.40e3T^{2} $
	11	$ 1 + 66.0iT - 1.46e4T^{2} $
	13	$ 1 + 143.T + 2.85e4T^{2} $
	17	$ 1 + 88.1iT - 8.35e4T^{2} $
	19	$ 1 - 397.T + 1.30e5T^{2} $
	23	$ 1 - 189. iT - 2.79e5T^{2} $
	29	$ 1 + 649. iT - 7.07e5T^{2} $
	31	$ 1 - 508.T + 9.23e5T^{2} $
	37	$ 1 - 1.27e3T + 1.87e6T^{2} $
	41	$ 1 - 1.53e3iT - 2.82e6T^{2} $
	43	$ 1 + 1.27e3T + 3.41e6T^{2} $
	47	$ 1 - 3.09e3iT - 4.87e6T^{2} $
	53	$ 1 - 940. iT - 7.89e6T^{2} $
	59	$ 1 - 5.07e3iT - 1.21e7T^{2} $
	61	$ 1 + 625.T + 1.38e7T^{2} $
	67	$ 1 + 2.76e3T + 2.01e7T^{2} $
	71	$ 1 + 2.56e3iT - 2.54e7T^{2} $
	73	$ 1 + 7.89e3T + 2.83e7T^{2} $
	79	$ 1 - 7.07e3T + 3.89e7T^{2} $
	83	$ 1 + 1.04e4iT - 4.74e7T^{2} $
	89	$ 1 - 1.40e4iT - 6.27e7T^{2} $
	97	$ 1 - 1.51e4T + 8.85e7T^{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

−11.60374504235181873451401294729, −10.43461735052508204981726208156, −9.666538034756457367322814210740, −9.186000577747940376660479214453, −7.76219121537830319575221122363, −5.95010569433798428787949889480, −4.62793436252922556572375477840, −3.34179027018983582040180205732, −2.65965062250797308199030004647, −0.896347951904850342127275710414, 0.30468788553446724205997237759, 3.41112438238144059783163340455, 4.73386848904615676562986279543, 5.78487688345651984851699374355, 6.78517738824805616263467851577, 7.36516760453908231208658618525, 8.573240437786316780945886689346, 9.604126602196345584733230734581, 10.04213689407154495974995872456, 12.26619016446529430740670984638

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(0.6396675728\)
\(L(\frac12)\)	\(\approx\)	\(0.6396675728\)
\(L(3)\)		not available
\(L(1)\)		not available

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