LMFDB - L-function 2-1900-95.94-c2-0-33

Dirichlet series

L(s) = 1

+ 0.284·3^-s + 2.19i·7^-s − 8.91·9^-s − 2.14·11^-s + 5.91·13^-s − 21.3i·17^-s + (12.8 + 13.9i)19^-s + 0.624i·21^-s + 9.06i·23^-s − 5.09·27^-s − 33.2i·29^-s + 44.1i·31^-s − 0.610·33^-s − 50.0·37^-s + 1.68·39^-s + ⋯

L(s) = 1

+ 0.0947·3^-s + 0.313i·7^-s − 0.991·9^-s − 0.195·11^-s + 0.454·13^-s − 1.25i·17^-s + (0.676 + 0.736i)19^-s + 0.0297i·21^-s + 0.393i·23^-s − 0.188·27^-s − 1.14i·29^-s + 1.42i·31^-s − 0.0185·33^-s − 1.35·37^-s + 0.0430·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.934 + 0.355i$
Analytic conductor:	$51.7712$
Root analytic conductor:	$7.19522$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (949, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1),\ 0.934 + 0.355i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.733577429$
$L(\frac12)$	$\approx$	$1.733577429$
$L(2)$		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 $
	19	$ 1 + (-12.8 - 13.9i)T $
good	3	$ 1 - 0.284T + 9T^{2} $
	7	$ 1 - 2.19iT - 49T^{2} $
	11	$ 1 + 2.14T + 121T^{2} $
	13	$ 1 - 5.91T + 169T^{2} $
	17	$ 1 + 21.3iT - 289T^{2} $
	23	$ 1 - 9.06iT - 529T^{2} $
	29	$ 1 + 33.2iT - 841T^{2} $
	31	$ 1 - 44.1iT - 961T^{2} $
	37	$ 1 + 50.0T + 1.36e3T^{2} $
	41	$ 1 + 61.1iT - 1.68e3T^{2} $
	43	$ 1 - 5.39iT - 1.84e3T^{2} $
	47	$ 1 - 9.02iT - 2.20e3T^{2} $
	53	$ 1 - 62.0T + 2.80e3T^{2} $
	59	$ 1 + 37.5iT - 3.48e3T^{2} $
	61	$ 1 - 58.0T + 3.72e3T^{2} $
	67	$ 1 - 121.T + 4.48e3T^{2} $
	71	$ 1 + 103. iT - 5.04e3T^{2} $
	73	$ 1 - 120. iT - 5.32e3T^{2} $
	79	$ 1 + 57.1iT - 6.24e3T^{2} $
	83	$ 1 - 47.3iT - 6.88e3T^{2} $
	89	$ 1 - 68.2iT - 7.92e3T^{2} $
	97	$ 1 + 22.7T + 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

−8.896120779534458149791856045842, −8.332797024858805675800857706167, −7.44524026977053802361771925023, −6.64747623933001376985113270830, −5.52864192878032023107865605939, −5.25849340273633080574155431170, −3.84707982436666073786026013239, −3.04566691421926870261758356127, −2.07811579829639254039173197222, −0.60985521070965332299048363992, 0.799705499545053930405581150758, 2.15030349278619386342802803705, 3.20154999760511336396501657839, 4.00528477780814875771578152032, 5.12769177320173140455225695583, 5.85500912515123967778036228860, 6.68784624669442087821923695385, 7.56067544363078842460360485884, 8.454235608474431148840207260962, 8.852625834695358568308669794721

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.733577429\)
\(L(\frac12)\)	\(\approx\)	\(1.733577429\)
\(L(2)\)		not available
\(L(1)\)		not available

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