LMFDB - L-function 2-1150-5.4-c3-0-22

Dirichlet series

L(s) = 1

− 2i·2^-s − 9i·3^-s − 4·4^-s − 18·6^-s − 2i·7^-s + 8i·8^-s − 54·9^-s − 52·11^-s + 36i·12^-s + 43i·13^-s − 4·14^-s + 16·16^-s + 50i·17^-s + 108i·18^-s + 74·19^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 1.73i·3^-s − 0.5·4^-s − 1.22·6^-s − 0.107i·7^-s + 0.353i·8^-s − 2·9^-s − 1.42·11^-s + 0.866i·12^-s + 0.917i·13^-s − 0.0763·14^-s + 0.250·16^-s + 0.713i·17^-s + 1.41i·18^-s + 0.893·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1150$ = $2 \cdot 5^{2} \cdot 23$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$67.8521$
Root analytic conductor:	$8.23724$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1150} (599, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1150,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.038582585$
$L(\frac12)$	$\approx$	$1.038582585$
$L(\frac{5}{2})$		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 + 2iT $
	5	$ 1 $
	23	$ 1 + 23iT $
good	3	$ 1 + 9iT - 27T^{2} $
	7	$ 1 + 2iT - 343T^{2} $
	11	$ 1 + 52T + 1.33e3T^{2} $
	13	$ 1 - 43iT - 2.19e3T^{2} $
	17	$ 1 - 50iT - 4.91e3T^{2} $
	19	$ 1 - 74T + 6.85e3T^{2} $
	29	$ 1 - 7T + 2.43e4T^{2} $
	31	$ 1 + 273T + 2.97e4T^{2} $
	37	$ 1 - 4iT - 5.06e4T^{2} $
	41	$ 1 - 123T + 6.89e4T^{2} $
	43	$ 1 + 152iT - 7.95e4T^{2} $
	47	$ 1 + 75iT - 1.03e5T^{2} $
	53	$ 1 - 86iT - 1.48e5T^{2} $
	59	$ 1 - 444T + 2.05e5T^{2} $
	61	$ 1 - 262T + 2.26e5T^{2} $
	67	$ 1 + 764iT - 3.00e5T^{2} $
	71	$ 1 + 21T + 3.57e5T^{2} $
	73	$ 1 - 681iT - 3.89e5T^{2} $
	79	$ 1 + 426T + 4.93e5T^{2} $
	83	$ 1 - 902iT - 5.71e5T^{2} $
	89	$ 1 - 1.27e3T + 7.04e5T^{2} $
	97	$ 1 - 342iT - 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

−9.156984753341019375458775190956, −8.353528440656486040837856379412, −7.59760117920956592063247197639, −6.99922095826349339610864908480, −5.91373870955692657670933847242, −5.15297251598224370797736254319, −3.71667334128060051893992150879, −2.53607321707441363466837332527, −1.87876833797541328977978690906, −0.78647153348708345755902956366, 0.33747226371362323416906897709, 2.73352320717575627650882667549, 3.51965193678172208993078727150, 4.59332813310046927712435113627, 5.42621783193972220427317116713, 5.63634894794154389426969570032, 7.25995950827715653244129532516, 7.978535356172127998203238028521, 8.874975414108071128362886110818, 9.570083636384678338223015620150

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.038582585\)
\(L(\frac12)\)	\(\approx\)	\(1.038582585\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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