LMFDB - L-function 2-1100-11.10-c2-0-34

Dirichlet series

L(s) = 1

− 3.11·3^-s − 12.3i·7^-s + 0.701·9^-s + (−8.70 − 6.72i)11^-s − 8.64i·13^-s − 12.3i·17^-s − 24.9i·19^-s + 38.3i·21^-s + 1.85·23^-s + 25.8·27^-s + 51.8i·29^-s − 22.5·31^-s + (27.1 + 20.9i)33^-s − 0.603·37^-s + 26.9i·39^-s + ⋯

L(s) = 1

− 1.03·3^-s − 1.75i·7^-s + 0.0779·9^-s + (−0.791 − 0.611i)11^-s − 0.664i·13^-s − 0.724i·17^-s − 1.31i·19^-s + 1.82i·21^-s + 0.0808·23^-s + 0.957·27^-s + 1.78i·29^-s − 0.726·31^-s + (0.821 + 0.635i)33^-s − 0.0163·37^-s + 0.690i·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1100$ = $2^{2} \cdot 5^{2} \cdot 11$
Sign:	$-0.791 - 0.611i$
Analytic conductor:	$29.9728$
Root analytic conductor:	$5.47474$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1100} (901, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1100,\ (\ :1),\ -0.791 - 0.611i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.4528029036$
$L(\frac12)$	$\approx$	$0.4528029036$
$L(2)$		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 $
	11	$ 1 + (8.70 + 6.72i)T $
good	3	$ 1 + 3.11T + 9T^{2} $
	7	$ 1 + 12.3iT - 49T^{2} $
	13	$ 1 + 8.64iT - 169T^{2} $
	17	$ 1 + 12.3iT - 289T^{2} $
	19	$ 1 + 24.9iT - 361T^{2} $
	23	$ 1 - 1.85T + 529T^{2} $
	29	$ 1 - 51.8iT - 841T^{2} $
	31	$ 1 + 22.5T + 961T^{2} $
	37	$ 1 + 0.603T + 1.36e3T^{2} $
	41	$ 1 + 49.8iT - 1.68e3T^{2} $
	43	$ 1 + 50.5iT - 1.84e3T^{2} $
	47	$ 1 - 31.1T + 2.20e3T^{2} $
	53	$ 1 - 51.7T + 2.80e3T^{2} $
	59	$ 1 + 35.1T + 3.48e3T^{2} $
	61	$ 1 - 2.00iT - 3.72e3T^{2} $
	67	$ 1 + 7.76T + 4.48e3T^{2} $
	71	$ 1 - 98.1T + 5.04e3T^{2} $
	73	$ 1 - 18.5iT - 5.32e3T^{2} $
	79	$ 1 - 130. iT - 6.24e3T^{2} $
	83	$ 1 + 23.3iT - 6.88e3T^{2} $
	89	$ 1 + 121.T + 7.92e3T^{2} $
	97	$ 1 + 99.1T + 9.40e3T^{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

−9.239141762522290767399122475874, −8.250308196963274367832949969992, −7.13634697798090947926856200469, −6.90954575886964082069055964214, −5.49215512035549079648133818651, −5.08526351698008020490401384425, −3.90537166048659919603840418385, −2.84796999519312341851622176049, −0.902356314465421248930010360729, −0.20315285834110559862551150799, 1.78816947295396462928959543586, 2.75651990993038894504721489996, 4.27370358658745187765420553967, 5.28366070802261717250419400476, 5.88562077376523949644142091396, 6.43161335451513219641201727365, 7.79621334368746437241609217857, 8.470214986510458624722617712263, 9.447551079142819794460962036292, 10.14072834924662702559117929054

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.4528029036\)
\(L(\frac12)\)	\(\approx\)	\(0.4528029036\)
\(L(2)\)		not available
\(L(1)\)		not available

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