LMFDB - L-function 2-75-5.4-c17-0-14

Dirichlet series

L(s) = 1

− 204i·2^-s − 6.56e3i·3^-s + 8.94e4·4^-s − 1.33e6·6^-s + 2.08e7i·7^-s − 4.49e7i·8^-s − 4.30e7·9^-s + 8.17e8·11^-s − 5.86e8i·12^-s + 2.99e8i·13^-s + 4.25e9·14^-s + 2.54e9·16^-s + 4.47e10i·17^-s + 8.78e9i·18^-s − 7.87e10·19^-s + ⋯

L(s) = 1

− 0.563i·2^-s − 0.577i·3^-s + 0.682·4^-s − 0.325·6^-s + 1.36i·7^-s − 0.948i·8^-s − 0.333·9^-s + 1.14·11^-s − 0.394i·12^-s + 0.101i·13^-s + 0.770·14^-s + 0.148·16^-s + 1.55i·17^-s + 0.187i·18^-s − 1.06·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$137.416$
Root analytic conductor:	$11.7224$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :17/2),\ 0.894 - 0.447i)$

Particular Values

$L(9)$	$\approx$	$2.456944028$
$L(\frac12)$	$\approx$	$2.456944028$
$L(\frac{19}{2})$		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 + 6.56e3iT $
	5	$ 1 $
good	2	$ 1 + 204iT - 1.31e5T^{2} $
	7	$ 1 - 2.08e7iT - 2.32e14T^{2} $
	11	$ 1 - 8.17e8T + 5.05e17T^{2} $
	13	$ 1 - 2.99e8iT - 8.65e18T^{2} $
	17	$ 1 - 4.47e10iT - 8.27e20T^{2} $
	19	$ 1 + 7.87e10T + 5.48e21T^{2} $
	23	$ 1 + 7.04e11iT - 1.41e23T^{2} $
	29	$ 1 - 1.63e11T + 7.25e24T^{2} $
	31	$ 1 - 1.04e12T + 2.25e25T^{2} $
	37	$ 1 - 1.98e13iT - 4.56e26T^{2} $
	41	$ 1 - 1.46e13T + 2.61e27T^{2} $
	43	$ 1 - 1.16e14iT - 5.87e27T^{2} $
	47	$ 1 - 1.76e14iT - 2.66e28T^{2} $
	53	$ 1 - 1.52e14iT - 2.05e29T^{2} $
	59	$ 1 - 2.62e14T + 1.27e30T^{2} $
	61	$ 1 + 1.35e15T + 2.24e30T^{2} $
	67	$ 1 + 4.44e14iT - 1.10e31T^{2} $
	71	$ 1 + 4.00e15T + 2.96e31T^{2} $
	73	$ 1 - 9.24e14iT - 4.74e31T^{2} $
	79	$ 1 + 1.47e16T + 1.81e32T^{2} $
	83	$ 1 - 2.64e16iT - 4.21e32T^{2} $
	89	$ 1 - 3.88e16T + 1.37e33T^{2} $
	97	$ 1 - 2.53e16iT - 5.95e33T^{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

−11.46735665598836191827545125566, −10.44424031819969694879704593106, −9.044562874080967407568164133649, −8.146228592766823899789361397380, −6.42007161415450355825367879513, −6.21265279140993258692000369224, −4.26736027408194011908559273156, −2.85838268029386630981386853314, −2.02335374033699888669899545071, −1.15703157203973959621220697005, 0.46724883501251450745307357119, 1.77163361199590327775268198971, 3.29701269746200863533873142733, 4.31282194664156559261003272020, 5.60369218538587544543227971521, 6.87239948351460227256103591019, 7.48564641194408385296880256738, 8.937347728497375908347526057777, 10.09756288621482758798421592083, 11.10902054119313580202795454202

Overview	Random
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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

\(L(9)\)	\(\approx\)	\(2.456944028\)
\(L(\frac12)\)	\(\approx\)	\(2.456944028\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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