LMFDB - L-function 2-1815-5.4-c1-0-52

Dirichlet series

L(s) = 1

− 0.311i·2^-s − i·3^-s + 1.90·4^-s + (0.311 + 2.21i)5^-s − 0.311·6^-s + 0.903i·7^-s − 1.21i·8^-s − 9^-s + (0.688 − 0.0967i)10^-s − 1.90i·12^-s + 2.90i·13^-s + 0.280·14^-s + (2.21 − 0.311i)15^-s + 3.42·16^-s − 2.28i·17^-s + 0.311i·18^-s + ⋯

L(s) = 1

− 0.219i·2^-s − 0.577i·3^-s + 0.951·4^-s + (0.139 + 0.990i)5^-s − 0.127·6^-s + 0.341i·7^-s − 0.429i·8^-s − 0.333·9^-s + (0.217 − 0.0306i)10^-s − 0.549i·12^-s + 0.805i·13^-s + 0.0750·14^-s + (0.571 − 0.0803i)15^-s + 0.857·16^-s − 0.553i·17^-s + 0.0733i·18^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1815$ = $3 \cdot 5 \cdot 11^{2}$
Sign:	$0.990 - 0.139i$
Analytic conductor:	$14.4928$
Root analytic conductor:	$3.80694$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1815} (364, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1815,\ (\ :1/2),\ 0.990 - 0.139i)$

Particular Values

$L(1)$	$\approx$	$2.315295302$
$L(\frac12)$	$\approx$	$2.315295302$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 + iT $
	5	$ 1 + (-0.311 - 2.21i)T $
	11	$ 1 $
good	2	$ 1 + 0.311iT - 2T^{2} $
	7	$ 1 - 0.903iT - 7T^{2} $
	13	$ 1 - 2.90iT - 13T^{2} $
	17	$ 1 + 2.28iT - 17T^{2} $
	19	$ 1 - 2.42T + 19T^{2} $
	23	$ 1 + 4iT - 23T^{2} $
	29	$ 1 - 7.05T + 29T^{2} $
	31	$ 1 + 2.62T + 31T^{2} $
	37	$ 1 - 5.80iT - 37T^{2} $
	41	$ 1 - 10.6T + 41T^{2} $
	43	$ 1 - 10.7iT - 43T^{2} $
	47	$ 1 - 0.949iT - 47T^{2} $
	53	$ 1 + 0.815iT - 53T^{2} $
	59	$ 1 - 1.67T + 59T^{2} $
	61	$ 1 - 7.24T + 61T^{2} $
	67	$ 1 - 12.8iT - 67T^{2} $
	71	$ 1 - 9.28T + 71T^{2} $
	73	$ 1 + 5.65iT - 73T^{2} $
	79	$ 1 + 16.5T + 79T^{2} $
	83	$ 1 - 7.76iT - 83T^{2} $
	89	$ 1 + 6.13T + 89T^{2} $
	97	$ 1 + 12.4iT - 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

−9.439486834820083013418961266625, −8.334398898374338595620680350932, −7.51196942092256612707064862350, −6.80921669318410880278476665727, −6.38210719544764168667270829710, −5.50123351535835164207195745482, −4.14612866525848453477311621689, −2.83760228211389467372013988638, −2.52800442751460155451008300145, −1.27605481207786890217452675056, 0.948841126212242353652141688304, 2.23111549249037264325708127808, 3.40113394879102789326935704875, 4.29239350833698380748839916308, 5.49381631171296279401457764767, 5.70352221820449873282007848249, 6.93445570551004657000911142847, 7.74080468678498988689764319293, 8.394449116531442432887036084465, 9.245897983719784020788316786545

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.315295302\)
\(L(\frac12)\)	\(\approx\)	\(2.315295302\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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