LMFDB - L-function 2-3150-21.20-c1-0-21

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s + (−0.0951 − 2.64i)7^-s + i·8^-s + 5.28i·11^-s + 2.19i·13^-s + (−2.64 + 0.0951i)14^-s + 16^-s + 1.04·17^-s − 6.43i·19^-s + 5.28·22^-s − 7.47i·23^-s + 2.19·26^-s + (0.0951 + 2.64i)28^-s + 7.47i·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + (−0.0359 − 0.999i)7^-s + 0.353i·8^-s + 1.59i·11^-s + 0.607i·13^-s + (−0.706 + 0.0254i)14^-s + 0.250·16^-s + 0.253·17^-s − 1.47i·19^-s + 1.12·22^-s − 1.55i·23^-s + 0.429·26^-s + (0.0179 + 0.499i)28^-s + 1.38i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3150$ = $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$
Sign:	$0.606 + 0.795i$
Analytic conductor:	$25.1528$
Root analytic conductor:	$5.01526$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3150} (251, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3150,\ (\ :1/2),\ 0.606 + 0.795i)$

Particular Values

$L(1)$	$\approx$	$1.654328877$
$L(\frac12)$	$\approx$	$1.654328877$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 + iT $
	3	$ 1 $
	5	$ 1 $
	7	$ 1 + (0.0951 + 2.64i)T $
good	11	$ 1 - 5.28iT - 11T^{2} $
	13	$ 1 - 2.19iT - 13T^{2} $
	17	$ 1 - 1.04T + 17T^{2} $
	19	$ 1 + 6.43iT - 19T^{2} $
	23	$ 1 + 7.47iT - 23T^{2} $
	29	$ 1 - 7.47iT - 29T^{2} $
	31	$ 1 - 9.09iT - 31T^{2} $
	37	$ 1 - 0.855T + 37T^{2} $
	41	$ 1 - 2.19T + 41T^{2} $
	43	$ 1 - 0.954T + 43T^{2} $
	47	$ 1 - 11.0T + 47T^{2} $
	53	$ 1 - 3.09iT - 53T^{2} $
	59	$ 1 - 13.7T + 59T^{2} $
	61	$ 1 + 8.05iT - 61T^{2} $
	67	$ 1 + 5.33T + 67T^{2} $
	71	$ 1 + 6.43iT - 71T^{2} $
	73	$ 1 - 4.57iT - 73T^{2} $
	79	$ 1 - 15.6T + 79T^{2} $
	83	$ 1 - 4.38T + 83T^{2} $
	89	$ 1 - 4.28T + 89T^{2} $
	97	$ 1 + 11.8iT - 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

−8.889536001444118320997211801045, −7.77530357548958237416842756072, −6.96276739088894987100586324320, −6.66623547711522328891786101228, −5.07867651789486836340328083791, −4.63668772444593705988499144979, −3.90869615286768573739352947181, −2.82855314870373420254563884319, −1.91373628530932425692672009455, −0.792558116109955453680233835246, 0.77581256488359184514870544512, 2.27895281693706704403938940712, 3.37121842645654013208745031075, 4.03339817805956161061747177144, 5.45011744319598491391844673049, 5.76889218952943858681398199057, 6.19532090761139222192136181494, 7.53633446381003833621592119265, 8.010107162426728118763372848706, 8.603525881222657457697479840190

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.654328877\)
\(L(\frac12)\)	\(\approx\)	\(1.654328877\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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