LMFDB - L-function 2-1450-29.28-c1-0-47

Dirichlet series

L(s) = 1

− i·2^-s − 2.09i·3^-s − 4^-s − 2.09·6^-s − 2.45·7^-s + i·8^-s − 1.36·9^-s − 5.50i·11^-s + 2.09i·12^-s − 3.41·13^-s + 2.45i·14^-s + 16^-s + 4.59i·17^-s + 1.36i·18^-s + 0.0902i·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 1.20i·3^-s − 0.5·4^-s − 0.853·6^-s − 0.929·7^-s + 0.353i·8^-s − 0.456·9^-s − 1.66i·11^-s + 0.603i·12^-s − 0.948·13^-s + 0.657i·14^-s + 0.250·16^-s + 1.11i·17^-s + 0.322i·18^-s + 0.0206i·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1450$ = $2 \cdot 5^{2} \cdot 29$
Sign:	$0.296 - 0.955i$
Analytic conductor:	$11.5783$
Root analytic conductor:	$3.40269$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1450} (1101, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1450,\ (\ :1/2),\ 0.296 - 0.955i)$

Particular Values

$L(1)$	$\approx$	$0.2486467448$
$L(\frac12)$	$\approx$	$0.2486467448$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	29	$ 1 + (-5.14 - 1.59i)T $
good	3	$ 1 + 2.09iT - 3T^{2} $
	7	$ 1 + 2.45T + 7T^{2} $
	11	$ 1 + 5.50iT - 11T^{2} $
	13	$ 1 + 3.41T + 13T^{2} $
	17	$ 1 - 4.59iT - 17T^{2} $
	19	$ 1 - 0.0902iT - 19T^{2} $
	23	$ 1 + 7.64T + 23T^{2} $
	31	$ 1 + 3.36iT - 31T^{2} $
	37	$ 1 - 9.01iT - 37T^{2} $
	41	$ 1 - 1.13iT - 41T^{2} $
	43	$ 1 + 4.91iT - 43T^{2} $
	47	$ 1 - 11.4iT - 47T^{2} $
	53	$ 1 - 8.41T + 53T^{2} $
	59	$ 1 + 5.04T + 59T^{2} $
	61	$ 1 - 4.96iT - 61T^{2} $
	67	$ 1 + 9.63T + 67T^{2} $
	71	$ 1 + 11.2T + 71T^{2} $
	73	$ 1 + 3.78iT - 73T^{2} $
	79	$ 1 - 1.22iT - 79T^{2} $
	83	$ 1 + 4.89T + 83T^{2} $
	89	$ 1 - 3.96iT - 89T^{2} $
	97	$ 1 + 4.98iT - 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

−8.747577019452999285536219403467, −8.166242837964020631770722081188, −7.34134859020075731060103205715, −6.15074986060750238664303129665, −6.01955143587205866826191904039, −4.43956169321069416597989175690, −3.35217407266434228265742816375, −2.53787030466845616364997574314, −1.34369735501406083400899682103, −0.10082159350828579879124744759, 2.33548619607475505407314818767, 3.58004682232832623322592283514, 4.50483192600372825102431855521, 4.96361591044031445220420812860, 6.03029666063806251853159747203, 7.08439826535937601412030529972, 7.47761208148008415064318061699, 8.790229064063495709467044654775, 9.526273756266434659303721515577, 10.00570210380077906888521100015

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.2486467448\)
\(L(\frac12)\)	\(\approx\)	\(0.2486467448\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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