LMFDB - L-function 2-35e2-5.4-c1-0-54

Dirichlet series

L(s) = 1

− 2i·2^-s − 3i·3^-s − 2·4^-s − 6·6^-s − 6·9^-s + 11^-s + 6i·12^-s − 3i·13^-s − 4·16^-s − 3i·17^-s + 12i·18^-s − 6·19^-s − 2i·22^-s + 4i·23^-s − 6·26^-s + 9i·27^-s + ⋯

L(s) = 1

− 1.41i·2^-s − 1.73i·3^-s − 4^-s − 2.44·6^-s − 2·9^-s + 0.301·11^-s + 1.73i·12^-s − 0.832i·13^-s − 16^-s − 0.727i·17^-s + 2.82i·18^-s − 1.37·19^-s − 0.426i·22^-s + 0.834i·23^-s − 1.17·26^-s + 1.73i·27^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1225$ = $5^{2} \cdot 7^{2}$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$9.78167$
Root analytic conductor:	$3.12756$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1225} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1225,\ (\ :1/2),\ 0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.106960141$
$L(\frac12)$	$\approx$	$1.106960141$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	5	$ 1 $
	7	$ 1 $
good	2	$ 1 + 2iT - 2T^{2} $
	3	$ 1 + 3iT - 3T^{2} $
	11	$ 1 - T + 11T^{2} $
	13	$ 1 + 3iT - 13T^{2} $
	17	$ 1 + 3iT - 17T^{2} $
	19	$ 1 + 6T + 19T^{2} $
	23	$ 1 - 4iT - 23T^{2} $
	29	$ 1 - T + 29T^{2} $
	31	$ 1 - 6T + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 - 6T + 41T^{2} $
	43	$ 1 - 6iT - 43T^{2} $
	47	$ 1 + 9iT - 47T^{2} $
	53	$ 1 - 10iT - 53T^{2} $
	59	$ 1 - 6T + 59T^{2} $
	61	$ 1 + 61T^{2} $
	67	$ 1 + 14iT - 67T^{2} $
	71	$ 1 + 8T + 71T^{2} $
	73	$ 1 + 6iT - 73T^{2} $
	79	$ 1 - T + 79T^{2} $
	83	$ 1 + 12iT - 83T^{2} $
	89	$ 1 + 12T + 89T^{2} $
	97	$ 1 + 15iT - 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

−9.085504784644961547718922652756, −8.272399070086999977551976521347, −7.43577560651644979215676320213, −6.63178873660114148930195814362, −5.83252675623711996914976713221, −4.48903850175522538646267425137, −3.17735066507596015996458248545, −2.43512016953117764432733915791, −1.48660056166459849829665747529, −0.46938527523322474674787705411, 2.50310320603050479019867807668, 4.09470757528731052252719026599, 4.37519704590598782595100743958, 5.38477804555668812413122714224, 6.21518213007494545174547244155, 6.85921945277758575910407963885, 8.248460520576903744176641803403, 8.621561449982198021140467333045, 9.384659824447197349055232318318, 10.21431070114476790487046223626

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.106960141\)
\(L(\frac12)\)	\(\approx\)	\(1.106960141\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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