LMFDB - L-function 2-4012-17.16-c1-0-16

Dirichlet series

L(s) = 1

− 0.510i·3^-s − 1.04i·5^-s + 1.04i·7^-s + 2.73·9^-s + 5.26i·11^-s − 4.96·13^-s − 0.534·15^-s + (1.40 − 3.87i)17^-s + 0.610·19^-s + 0.535·21^-s + 3.54i·23^-s + 3.90·25^-s − 2.92i·27^-s − 3.22i·29^-s + 4.83i·31^-s + ⋯

L(s) = 1

− 0.294i·3^-s − 0.468i·5^-s + 0.396i·7^-s + 0.913·9^-s + 1.58i·11^-s − 1.37·13^-s − 0.138·15^-s + (0.340 − 0.940i)17^-s + 0.139·19^-s + 0.116·21^-s + 0.739i·23^-s + 0.780·25^-s − 0.563i·27^-s − 0.599i·29^-s + 0.868i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4012$ = $2^{2} \cdot 17 \cdot 59$
Sign:	$0.340 - 0.940i$
Analytic conductor:	$32.0359$
Root analytic conductor:	$5.66003$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4012} (237, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4012,\ (\ :1/2),\ 0.340 - 0.940i)$

Particular Values

$L(1)$	$\approx$	$1.507528243$
$L(\frac12)$	$\approx$	$1.507528243$
$L(\frac{3}{2})$		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 $
	17	$ 1 + (-1.40 + 3.87i)T $
	59	$ 1 + T $
good	3	$ 1 + 0.510iT - 3T^{2} $
	5	$ 1 + 1.04iT - 5T^{2} $
	7	$ 1 - 1.04iT - 7T^{2} $
	11	$ 1 - 5.26iT - 11T^{2} $
	13	$ 1 + 4.96T + 13T^{2} $
	19	$ 1 - 0.610T + 19T^{2} $
	23	$ 1 - 3.54iT - 23T^{2} $
	29	$ 1 + 3.22iT - 29T^{2} $
	31	$ 1 - 4.83iT - 31T^{2} $
	37	$ 1 + 2.92iT - 37T^{2} $
	41	$ 1 - 9.92iT - 41T^{2} $
	43	$ 1 + 9.74T + 43T^{2} $
	47	$ 1 + 4.95T + 47T^{2} $
	53	$ 1 - 4.19T + 53T^{2} $
	61	$ 1 - 7.86iT - 61T^{2} $
	67	$ 1 + 10.1T + 67T^{2} $
	71	$ 1 - 4.35iT - 71T^{2} $
	73	$ 1 - 9.61iT - 73T^{2} $
	79	$ 1 - 5.29iT - 79T^{2} $
	83	$ 1 + 10.5T + 83T^{2} $
	89	$ 1 - 11.7T + 89T^{2} $
	97	$ 1 - 8.88iT - 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

−8.582585551655615386638208005217, −7.60414406141699867836145088165, −7.23343041740136820144642594967, −6.64762298648256264775777449208, −5.37535209566272624104780492354, −4.84701008255533544142380654388, −4.28550112196893216745319900801, −2.94908026502400880263887580752, −2.09627042268810126734017302068, −1.18721114871424276521406842552, 0.45526424455614749088106376805, 1.76367045433035500403638307867, 2.96108949898191527278833205574, 3.60908760297041711515409017458, 4.49535322386120552626897303077, 5.25964175132952062272280604225, 6.16206430841243610400061552454, 6.88456093199786635856549469801, 7.50777728026939196957150919096, 8.302964559909368110464187349508

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.507528243\)
\(L(\frac12)\)	\(\approx\)	\(1.507528243\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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