LMFDB - L-function 2-2300-5.4-c3-0-68

Dirichlet series

L(s) = 1

− 6.13i·3^-s + 19.8i·7^-s − 10.6·9^-s − 55.2·11^-s − 1.83i·13^-s + 130. i·17^-s + 17.3·19^-s + 121.·21^-s − 23i·23^-s − 100. i·27^-s − 77.6·29^-s + 206.·31^-s + 338. i·33^-s + 251. i·37^-s − 11.2·39^-s + ⋯

L(s) = 1

− 1.18i·3^-s + 1.07i·7^-s − 0.393·9^-s − 1.51·11^-s − 0.0391i·13^-s + 1.85i·17^-s + 0.208·19^-s + 1.26·21^-s − 0.208i·23^-s − 0.716i·27^-s − 0.497·29^-s + 1.19·31^-s + 1.78i·33^-s + 1.11i·37^-s − 0.0462·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$135.704$
Root analytic conductor:	$11.6492$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1749, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :3/2),\ -0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.6886617845$
$L(\frac12)$	$\approx$	$0.6886617845$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
	23	$ 1 + 23iT $
good	3	$ 1 + 6.13iT - 27T^{2} $
	7	$ 1 - 19.8iT - 343T^{2} $
	11	$ 1 + 55.2T + 1.33e3T^{2} $
	13	$ 1 + 1.83iT - 2.19e3T^{2} $
	17	$ 1 - 130. iT - 4.91e3T^{2} $
	19	$ 1 - 17.3T + 6.85e3T^{2} $
	29	$ 1 + 77.6T + 2.43e4T^{2} $
	31	$ 1 - 206.T + 2.97e4T^{2} $
	37	$ 1 - 251. iT - 5.06e4T^{2} $
	41	$ 1 + 157.T + 6.89e4T^{2} $
	43	$ 1 + 336. iT - 7.95e4T^{2} $
	47	$ 1 + 488. iT - 1.03e5T^{2} $
	53	$ 1 - 562. iT - 1.48e5T^{2} $
	59	$ 1 - 125.T + 2.05e5T^{2} $
	61	$ 1 - 163.T + 2.26e5T^{2} $
	67	$ 1 - 769. iT - 3.00e5T^{2} $
	71	$ 1 - 113.T + 3.57e5T^{2} $
	73	$ 1 + 747. iT - 3.89e5T^{2} $
	79	$ 1 + 1.11e3T + 4.93e5T^{2} $
	83	$ 1 + 988. iT - 5.71e5T^{2} $
	89	$ 1 + 1.38e3T + 7.04e5T^{2} $
	97	$ 1 + 1.18e3iT - 9.12e5T^{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

−8.444223023202325227100167103743, −7.59260272592863920164433573045, −6.82213457246718184980900734005, −5.93178711732576304261908796666, −5.49339312724569094121746327186, −4.35330789641621675384585804859, −3.04305927958178255516714991471, −2.25375830587764812679930052808, −1.50577018348259849735334198079, −0.15591038483579904556951374453, 0.906848964720949155733303957375, 2.54198203534351246946011030540, 3.32846622499417302003776750030, 4.28490637912268154095380403012, 4.91743330435719334370339148784, 5.50318170792979599391928552660, 6.81905397323470392142373618077, 7.49641928631971404190523937955, 8.142657893048373619264002688765, 9.287447061589371108423573320094

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

\(L(2)\)	\(\approx\)	\(0.6886617845\)
\(L(\frac12)\)	\(\approx\)	\(0.6886617845\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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