LMFDB - L-function 2-35e2-5.4-c1-0-2

Dirichlet series

L(s) = 1

+ 2.19i·2^-s + 2.83i·3^-s − 2.83·4^-s − 6.23·6^-s − 1.83i·8^-s − 5.03·9^-s − 2.56·11^-s − 8.03i·12^-s − 0.563i·13^-s − 1.63·16^-s − 5.19i·17^-s − 11.0i·18^-s − 0.469·19^-s − 5.63i·22^-s + 4.03i·23^-s + 5.19·24^-s + ⋯

L(s) = 1

+ 1.55i·2^-s + 1.63i·3^-s − 1.41·4^-s − 2.54·6^-s − 0.648i·8^-s − 1.67·9^-s − 0.772·11^-s − 2.31i·12^-s − 0.156i·13^-s − 0.408·16^-s − 1.26i·17^-s − 2.60i·18^-s − 0.107·19^-s − 1.20i·22^-s + 0.840i·23^-s + 1.06·24^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1225$ = $5^{2} \cdot 7^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$9.78167$
Root analytic conductor:	$3.12756$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1225} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1225,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.5424403716$
$L(\frac12)$	$\approx$	$0.5424403716$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	5	$ 1 $
	7	$ 1 $
good	2	$ 1 - 2.19iT - 2T^{2} $
	3	$ 1 - 2.83iT - 3T^{2} $
	11	$ 1 + 2.56T + 11T^{2} $
	13	$ 1 + 0.563iT - 13T^{2} $
	17	$ 1 + 5.19iT - 17T^{2} $
	19	$ 1 + 0.469T + 19T^{2} $
	23	$ 1 - 4.03iT - 23T^{2} $
	29	$ 1 + 2.86T + 29T^{2} $
	31	$ 1 + 3.86T + 31T^{2} $
	37	$ 1 + 2.23iT - 37T^{2} $
	41	$ 1 - 3.30T + 41T^{2} $
	43	$ 1 - 10.4iT - 43T^{2} $
	47	$ 1 + 9.36iT - 47T^{2} $
	53	$ 1 - 10.7iT - 53T^{2} $
	59	$ 1 + 3.96T + 59T^{2} $
	61	$ 1 + 3.86T + 61T^{2} $
	67	$ 1 - 7.66iT - 67T^{2} $
	71	$ 1 + 8.73T + 71T^{2} $
	73	$ 1 + 11.3iT - 73T^{2} $
	79	$ 1 + 5.15T + 79T^{2} $
	83	$ 1 - 3.13iT - 83T^{2} $
	89	$ 1 - 5.57T + 89T^{2} $
	97	$ 1 - 15.2iT - 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

−10.23759888446192883194595417942, −9.287459881822234424156744188537, −9.043480979777601312590734078091, −7.87831908069479668301654239374, −7.34422349375751167179908345984, −6.11702070963696177503107837414, −5.33356741536742830341891557362, −4.87309836274324346245671461791, −3.92659211226604239589183062009, −2.77660491898724571109635759744, 0.22337462666274272724815398284, 1.54202970312033006529092417439, 2.21695896597398907907171267376, 3.16302251624480757608297441748, 4.31245432492963397199706710615, 5.61833406843256982959204422485, 6.53120008834170929742918553200, 7.41320471294350118688448272758, 8.268299089496267951256025609995, 8.947475563729411109692923113887

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5424403716\)
\(L(\frac12)\)	\(\approx\)	\(0.5424403716\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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