LMFDB - L-function 2-15e2-15.14-c2-0-0

Dirichlet series

L(s) = 1

− 1.41·2^-s − 1.99·4^-s − 9i·7^-s + 8.48·8^-s + 18.3i·11^-s − i·13^-s + 12.7i·14^-s − 4.00·16^-s − 26.8·17^-s − 25·19^-s − 26i·22^-s − 4.24·23^-s + 1.41i·26^-s + 17.9i·28^-s + 26.8i·29^-s + ⋯

L(s) = 1

− 0.707·2^-s − 0.499·4^-s − 1.28i·7^-s + 1.06·8^-s + 1.67i·11^-s − 0.0769i·13^-s + 0.909i·14^-s − 0.250·16^-s − 1.58·17^-s − 1.31·19^-s − 1.18i·22^-s − 0.184·23^-s + 0.0543i·26^-s + 0.642i·28^-s + 0.926i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$-0.881 - 0.472i$
Analytic conductor:	$6.13080$
Root analytic conductor:	$2.47604$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :1),\ -0.881 - 0.472i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.0348241 + 0.138791i$
$L(\frac12)$	$\approx$	$0.0348241 + 0.138791i$
$L(2)$		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 + 1.41T + 4T^{2} $
	7	$ 1 + 9iT - 49T^{2} $
	11	$ 1 - 18.3iT - 121T^{2} $
	13	$ 1 + iT - 169T^{2} $
	17	$ 1 + 26.8T + 289T^{2} $
	19	$ 1 + 25T + 361T^{2} $
	23	$ 1 + 4.24T + 529T^{2} $
	29	$ 1 - 26.8iT - 841T^{2} $
	31	$ 1 + 39T + 961T^{2} $
	37	$ 1 - 32iT - 1.36e3T^{2} $
	41	$ 1 + 5.65iT - 1.68e3T^{2} $
	43	$ 1 - 23iT - 1.84e3T^{2} $
	47	$ 1 - 32.5T + 2.20e3T^{2} $
	53	$ 1 + 96.1T + 2.80e3T^{2} $
	59	$ 1 + 9.89iT - 3.48e3T^{2} $
	61	$ 1 - 73T + 3.72e3T^{2} $
	67	$ 1 - 63iT - 4.48e3T^{2} $
	71	$ 1 - 62.2iT - 5.04e3T^{2} $
	73	$ 1 + 136iT - 5.32e3T^{2} $
	79	$ 1 - 24T + 6.24e3T^{2} $
	83	$ 1 + 46.6T + 6.88e3T^{2} $
	89	$ 1 + 101. iT - 7.92e3T^{2} $
	97	$ 1 + 7iT - 9.40e3T^{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

−12.69148457648470224551471994841, −11.03973111890634842038801481293, −10.39172533083587078047896495486, −9.547423903321555688602969509668, −8.586475717743626918410703000928, −7.45107828840233738354518272851, −6.75792049500721193427429627186, −4.73716502826992248513158133671, −4.11453094233050258327389252582, −1.77966314376157158665249237127, 0.097613318755117268343618852871, 2.24414519363540639149870404666, 4.01401840891701747357018558550, 5.42706721921762772217013368174, 6.43723931929326329169989754538, 8.067998430976131239372835984718, 8.794217008802911550194910065482, 9.257231189911555147536538886510, 10.71533300186033744420473699982, 11.31644444653602119121844160420

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.0348241 + 0.138791i\)
\(L(\frac12)\)	\(\approx\)	\(0.0348241 + 0.138791i\)
\(L(2)\)		not available
\(L(1)\)		not available

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