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

Dirichlet series

L(s) = 1

− 1.41i·2^-s − 0.414i·3^-s − 0.585·6^-s − 2.82i·8^-s + 2.82·9^-s − 0.171·11^-s + 4.41i·13^-s − 4.00·16^-s − 3.24i·17^-s − 4.00i·18^-s + 6·19^-s + 0.242i·22^-s − 7.41i·23^-s − 1.17·24^-s + 6.24·26^-s − 2.41i·27^-s + ⋯

L(s) = 1

− 0.999i·2^-s − 0.239i·3^-s − 0.239·6^-s − 0.999i·8^-s + 0.942·9^-s − 0.0517·11^-s + 1.22i·13^-s − 1.00·16^-s − 0.786i·17^-s − 0.942i·18^-s + 1.37·19^-s + 0.0517i·22^-s − 1.54i·23^-s − 0.239·24^-s + 1.22·26^-s − 0.464i·27^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1225$ = $5^{2} \cdot 7^{2}$
Sign:	$-0.447 + 0.894i$
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:	$(2,\ 1225,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.009498165$
$L(\frac12)$	$\approx$	$2.009498165$
$L(\frac{3}{2})$		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 + 1.41iT - 2T^{2} $
	3	$ 1 + 0.414iT - 3T^{2} $
	11	$ 1 + 0.171T + 11T^{2} $
	13	$ 1 - 4.41iT - 13T^{2} $
	17	$ 1 + 3.24iT - 17T^{2} $
	19	$ 1 - 6T + 19T^{2} $
	23	$ 1 + 7.41iT - 23T^{2} $
	29	$ 1 - 8.65T + 29T^{2} $
	31	$ 1 + 10.2T + 31T^{2} $
	37	$ 1 - 2.24iT - 37T^{2} $
	41	$ 1 - 6.24T + 41T^{2} $
	43	$ 1 + 2iT - 43T^{2} $
	47	$ 1 - 7.24iT - 47T^{2} $
	53	$ 1 + 4.24iT - 53T^{2} $
	59	$ 1 + 2.24T + 59T^{2} $
	61	$ 1 - 2.82T + 61T^{2} $
	67	$ 1 + 8.24iT - 67T^{2} $
	71	$ 1 + 3.17T + 71T^{2} $
	73	$ 1 + 8.48iT - 73T^{2} $
	79	$ 1 + 1.48T + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 8T + 89T^{2} $
	97	$ 1 + 13.2iT - 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

−9.588391877015130926403781554186, −9.012838989317254877815701031064, −7.67121075129378543780655603201, −6.98363290492746593740983918365, −6.34696800522362573593715694532, −4.87825936341457816741149423533, −4.10721730909854452609510069331, −3.00045444112875943221989240551, −2.01953311062371501653911428773, −0.959370353630057669573196307846, 1.44714350332016053173373185346, 2.94969784261575023663820918899, 4.02768448642237215175657012170, 5.32645511438736651570010808757, 5.64471692916712948508154018816, 6.84045061439501602714711681239, 7.49823009222978381648430830092, 8.040400528572469758229482029210, 9.072067502339248073432191906592, 9.937019805959793207517045237546

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.009498165\)
\(L(\frac12)\)	\(\approx\)	\(2.009498165\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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