LMFDB - L-function 2-1216-152.75-c1-0-6

Dirichlet series

L(s) = 1

+ 3.40i·3^-s + 4.62i·7^-s − 8.56·9^-s + 0.223·13^-s − 3.69·17^-s − 4.35i·19^-s − 15.7·21^-s + 6.49i·23^-s + 5·25^-s − 18.9i·27^-s + 6.57·29^-s − 8.71·37^-s + 0.761i·39^-s + 6i·47^-s − 14.4·49^-s + ⋯

L(s) = 1

+ 1.96i·3^-s + 1.74i·7^-s − 2.85·9^-s + 0.0620·13^-s − 0.896·17^-s − 0.999i·19^-s − 3.43·21^-s + 1.35i·23^-s + 25^-s − 3.64i·27^-s + 1.22·29^-s − 1.43·37^-s + 0.121i·39^-s + 0.875i·47^-s − 2.06·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$9.70980$
Root analytic conductor:	$3.11605$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1/2),\ -0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.072439141$
$L(\frac12)$	$\approx$	$1.072439141$
$L(\frac{3}{2})$		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 $
	19	$ 1 + 4.35iT $
good	3	$ 1 - 3.40iT - 3T^{2} $
	5	$ 1 - 5T^{2} $
	7	$ 1 - 4.62iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 - 0.223T + 13T^{2} $
	17	$ 1 + 3.69T + 17T^{2} $
	23	$ 1 - 6.49iT - 23T^{2} $
	29	$ 1 - 6.57T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 8.71T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 - 13.8T + 53T^{2} $
	59	$ 1 + 2.95iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 10.6iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 1.82T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 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

−10.15053444550387956077660807171, −9.280111750085338042551482640704, −8.891186162271538954552144852330, −8.359052510903860639203916533728, −6.70413436449514002925873041512, −5.63329201267292235156841779203, −5.15907698055409035076454770711, −4.33983404975611072330396256558, −3.16629192483419579908779619337, −2.48814582241437693558831417997, 0.46324710390559103497528091274, 1.42381966829404287055727586117, 2.59854784271875324489770152520, 3.81352528800858615301340932305, 5.05089604853988102654477194056, 6.38124772592024266158399393760, 6.77962087857234738656363934211, 7.40664522805829684594495614224, 8.248610021536731023210375520929, 8.797101312446899810861241068814

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.072439141\)
\(L(\frac12)\)	\(\approx\)	\(1.072439141\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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