LMFDB - L-function 2-60e2-12.11-c1-0-28

Dirichlet series

L(s) = 1

+ 4.24i·7^-s + 2.44·11^-s − 2.44·13^-s − 6.92i·17^-s − 6.92i·19^-s − 6·23^-s − 2.82i·29^-s + 3.46i·31^-s − 2.44·37^-s − 7.07i·41^-s − 8.48i·43^-s − 10.9·49^-s + 12.2·59^-s − 2·61^-s + 9.79·71^-s + ⋯

L(s) = 1

+ 1.60i·7^-s + 0.738·11^-s − 0.679·13^-s − 1.68i·17^-s − 1.58i·19^-s − 1.25·23^-s − 0.525i·29^-s + 0.622i·31^-s − 0.402·37^-s − 1.10i·41^-s − 1.29i·43^-s − 1.57·49^-s + 1.59·59^-s − 0.256·61^-s + 1.16·71^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3600$ = $2^{4} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.418 + 0.908i$
Analytic conductor:	$28.7461$
Root analytic conductor:	$5.36154$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3600} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3600,\ (\ :1/2),\ 0.418 + 0.908i)$

Particular Values

$L(1)$	$\approx$	$1.329812810$
$L(\frac12)$	$\approx$	$1.329812810$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 - 4.24iT - 7T^{2} $
	11	$ 1 - 2.44T + 11T^{2} $
	13	$ 1 + 2.44T + 13T^{2} $
	17	$ 1 + 6.92iT - 17T^{2} $
	19	$ 1 + 6.92iT - 19T^{2} $
	23	$ 1 + 6T + 23T^{2} $
	29	$ 1 + 2.82iT - 29T^{2} $
	31	$ 1 - 3.46iT - 31T^{2} $
	37	$ 1 + 2.44T + 37T^{2} $
	41	$ 1 + 7.07iT - 41T^{2} $
	43	$ 1 + 8.48iT - 43T^{2} $
	47	$ 1 + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 - 12.2T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 - 9.79T + 71T^{2} $
	73	$ 1 + 4.89T + 73T^{2} $
	79	$ 1 + 10.3iT - 79T^{2} $
	83	$ 1 - 12T + 83T^{2} $
	89	$ 1 + 7.07iT - 89T^{2} $
	97	$ 1 - 14.6T + 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

−8.721458300227838924523925927410, −7.60823508193316187776523529553, −6.92993172070863198315168571389, −6.19010682819049336862675298976, −5.25474836300376965596880159430, −4.87546903690457074846285397256, −3.67187975358911681336690803467, −2.58998062019123278262478033008, −2.13695032376327114616724135151, −0.41897365365765715642621457782, 1.13741299682006040934012819366, 1.96740256988932410747501470768, 3.55612311094825160268989884650, 3.92435948420777677409463011605, 4.67089400078051643251830049249, 5.90307227868478750697369666084, 6.42059132448223072246371916807, 7.27400154621157893316746287453, 7.967338945040718866582951540829, 8.424210135129440707845654474026

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.329812810\)
\(L(\frac12)\)	\(\approx\)	\(1.329812810\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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