LMFDB - L-function 2-75-5.4-c17-0-17

Dirichlet series

L(s) = 1

− 367. i·2^-s + 6.56e3i·3^-s − 3.77e3·4^-s + 2.40e6·6^-s − 1.91e7i·7^-s − 4.67e7i·8^-s − 4.30e7·9^-s − 1.08e9·11^-s − 2.47e7i·12^-s + 4.07e9i·13^-s − 7.02e9·14^-s − 1.76e10·16^-s − 1.03e9i·17^-s + 1.58e10i·18^-s + 8.27e10·19^-s + ⋯

L(s) = 1

− 1.01i·2^-s + 0.577i·3^-s − 0.0288·4^-s + 0.585·6^-s − 1.25i·7^-s − 0.985i·8^-s − 0.333·9^-s − 1.52·11^-s − 0.0166i·12^-s + 1.38i·13^-s − 1.27·14^-s − 1.02·16^-s − 0.0359i·17^-s + 0.338i·18^-s + 1.11·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$137.416$
Root analytic conductor:	$11.7224$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :17/2),\ 0.894 + 0.447i)$

Particular Values

$L(9)$	$\approx$	$1.853625467$
$L(\frac12)$	$\approx$	$1.853625467$
$L(\frac{19}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 - 6.56e3iT $
	5	$ 1 $
good	2	$ 1 + 367. iT - 1.31e5T^{2} $
	7	$ 1 + 1.91e7iT - 2.32e14T^{2} $
	11	$ 1 + 1.08e9T + 5.05e17T^{2} $
	13	$ 1 - 4.07e9iT - 8.65e18T^{2} $
	17	$ 1 + 1.03e9iT - 8.27e20T^{2} $
	19	$ 1 - 8.27e10T + 5.48e21T^{2} $
	23	$ 1 - 5.01e11iT - 1.41e23T^{2} $
	29	$ 1 + 3.86e12T + 7.25e24T^{2} $
	31	$ 1 - 8.19e12T + 2.25e25T^{2} $
	37	$ 1 - 1.23e13iT - 4.56e26T^{2} $
	41	$ 1 + 7.08e13T + 2.61e27T^{2} $
	43	$ 1 - 3.21e13iT - 5.87e27T^{2} $
	47	$ 1 - 1.79e13iT - 2.66e28T^{2} $
	53	$ 1 - 1.35e14iT - 2.05e29T^{2} $
	59	$ 1 - 1.26e15T + 1.27e30T^{2} $
	61	$ 1 - 2.30e15T + 2.24e30T^{2} $
	67	$ 1 + 4.62e15iT - 1.10e31T^{2} $
	71	$ 1 - 1.04e16T + 2.96e31T^{2} $
	73	$ 1 + 6.57e15iT - 4.74e31T^{2} $
	79	$ 1 + 7.36e15T + 1.81e32T^{2} $
	83	$ 1 - 6.77e15iT - 4.21e32T^{2} $
	89	$ 1 + 6.56e15T + 1.37e33T^{2} $
	97	$ 1 - 2.22e16iT - 5.95e33T^{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

−11.11337953875972668571660620019, −10.14341637481360951401743871306, −9.550348228551179674641408060760, −7.80724939178179303441352216272, −6.81039564621140868239436431180, −5.12803417062588703644224483172, −3.98572623434793826644174031359, −3.12991836878344683363255992524, −1.88538743438754045767936707652, −0.74954968257018501273065946316, 0.47616834394617611829573787102, 2.24872285872227428803530140194, 2.87572494090633726264944847592, 5.33546081143352208308075939690, 5.55548893961190702095443426511, 6.88588729885263950525869981339, 7.993509898294708350430005736092, 8.479155904301583288251405172228, 10.16043937104622222014631394378, 11.42346191026126742300392753793

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

\(L(9)\)	\(\approx\)	\(1.853625467\)
\(L(\frac12)\)	\(\approx\)	\(1.853625467\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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