LMFDB - L-function 2-525-35.34-c2-0-6

Dirichlet series

L(s) = 1

− 3.50i·2^-s + 1.73·3^-s − 8.27·4^-s − 6.06i·6^-s + (−2.03 + 6.69i)7^-s + 14.9i·8^-s + 2.99·9^-s − 2.03·11^-s − 14.3·12^-s − 18.0·13^-s + (23.4 + 7.13i)14^-s + 19.3·16^-s − 1.07·17^-s − 10.5i·18^-s + 28.7i·19^-s + ⋯

L(s) = 1

− 1.75i·2^-s + 0.577·3^-s − 2.06·4^-s − 1.01i·6^-s + (−0.290 + 0.956i)7^-s + 1.87i·8^-s + 0.333·9^-s − 0.184·11^-s − 1.19·12^-s − 1.38·13^-s + (1.67 + 0.509i)14^-s + 1.21·16^-s − 0.0631·17^-s − 0.583i·18^-s + 1.51i·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$525$ = $3 \cdot 5^{2} \cdot 7$
Sign:	$0.985 - 0.167i$
Analytic conductor:	$14.3052$
Root analytic conductor:	$3.78222$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{525} (349, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 525,\ (\ :1),\ 0.985 - 0.167i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9205666475$
$L(\frac12)$	$\approx$	$0.9205666475$
$L(2)$		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 - 1.73T $
	5	$ 1 $
	7	$ 1 + (2.03 - 6.69i)T $
good	2	$ 1 + 3.50iT - 4T^{2} $
	11	$ 1 + 2.03T + 121T^{2} $
	13	$ 1 + 18.0T + 169T^{2} $
	17	$ 1 + 1.07T + 289T^{2} $
	19	$ 1 - 28.7iT - 361T^{2} $
	23	$ 1 - 24.8iT - 529T^{2} $
	29	$ 1 - 38.4T + 841T^{2} $
	31	$ 1 - 44.0iT - 961T^{2} $
	37	$ 1 + 37.2iT - 1.36e3T^{2} $
	41	$ 1 + 49.9iT - 1.68e3T^{2} $
	43	$ 1 + 9.58iT - 1.84e3T^{2} $
	47	$ 1 + 55.6T + 2.20e3T^{2} $
	53	$ 1 - 57.4iT - 2.80e3T^{2} $
	59	$ 1 - 101. iT - 3.48e3T^{2} $
	61	$ 1 - 31.9iT - 3.72e3T^{2} $
	67	$ 1 + 95.7iT - 4.48e3T^{2} $
	71	$ 1 + 25.8T + 5.04e3T^{2} $
	73	$ 1 + 95.6T + 5.32e3T^{2} $
	79	$ 1 + 28.1T + 6.24e3T^{2} $
	83	$ 1 + 103.T + 6.88e3T^{2} $
	89	$ 1 - 29.3iT - 7.92e3T^{2} $
	97	$ 1 - 67.6T + 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

−10.48889625188076801376024243111, −9.981319905078342604729035705471, −9.176083728119942664486056336690, −8.475495799988149376861284462929, −7.31118460872341511062522585595, −5.66900720313126062045523432561, −4.62549512694097450261618405320, −3.41909427572020608603768646141, −2.62354625013037883505967073883, −1.63593699503033346962308085774, 0.33169338304231025547098285413, 2.81049250977634740576294273925, 4.43773074686547298543175228666, 4.88758397537552428409156710222, 6.45359144555413141883745848596, 6.95360705443822876605984921815, 7.80749441950230782286814053817, 8.479798092532841074810553204184, 9.579486238001574639417776874670, 10.08396837389056198229938427758

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9205666475\)
\(L(\frac12)\)	\(\approx\)	\(0.9205666475\)
\(L(2)\)		not available
\(L(1)\)		not available

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