LMFDB - L-function 2-234-13.12-c7-0-23

Dirichlet series

L(s) = 1

− 8i·2^-s − 64·4^-s − 319. i·5^-s − 1.00e3i·7^-s + 512i·8^-s − 2.55e3·10^-s + 7.26e3i·11^-s + (7.92e3 − 43.0i)13^-s − 8.06e3·14^-s + 4.09e3·16^-s + 2.54e4·17^-s − 8.74e3i·19^-s + 2.04e4i·20^-s + 5.80e4·22^-s + 8.35e4·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.14i·5^-s − 1.11i·7^-s + 0.353i·8^-s − 0.807·10^-s + 1.64i·11^-s + (0.999 − 0.00543i)13^-s − 0.785·14^-s + 0.250·16^-s + 1.25·17^-s − 0.292i·19^-s + 0.570i·20^-s + 1.16·22^-s + 1.43·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00543 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.00543 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$234$ = $2 \cdot 3^{2} \cdot 13$
Sign:	$0.00543 + 0.999i$
Analytic conductor:	$73.0980$
Root analytic conductor:	$8.54974$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{234} (181, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 234,\ (\ :7/2),\ 0.00543 + 0.999i)$

Particular Values

$L(4)$	$\approx$	$2.403930797$
$L(\frac12)$	$\approx$	$2.403930797$
$L(\frac{9}{2})$		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 + 8iT $
	3	$ 1 $
	13	$ 1 + (-7.92e3 + 43.0i)T $
good	5	$ 1 + 319. iT - 7.81e4T^{2} $
	7	$ 1 + 1.00e3iT - 8.23e5T^{2} $
	11	$ 1 - 7.26e3iT - 1.94e7T^{2} $
	17	$ 1 - 2.54e4T + 4.10e8T^{2} $
	19	$ 1 + 8.74e3iT - 8.93e8T^{2} $
	23	$ 1 - 8.35e4T + 3.40e9T^{2} $
	29	$ 1 - 1.97e5T + 1.72e10T^{2} $
	31	$ 1 - 1.38e5iT - 2.75e10T^{2} $
	37	$ 1 - 6.18e4iT - 9.49e10T^{2} $
	41	$ 1 - 7.84e5iT - 1.94e11T^{2} $
	43	$ 1 + 4.40e5T + 2.71e11T^{2} $
	47	$ 1 - 1.80e5iT - 5.06e11T^{2} $
	53	$ 1 - 5.94e5T + 1.17e12T^{2} $
	59	$ 1 - 4.34e4iT - 2.48e12T^{2} $
	61	$ 1 - 6.07e5T + 3.14e12T^{2} $
	67	$ 1 + 3.33e6iT - 6.06e12T^{2} $
	71	$ 1 + 3.42e6iT - 9.09e12T^{2} $
	73	$ 1 - 2.18e6iT - 1.10e13T^{2} $
	79	$ 1 + 4.16e6T + 1.92e13T^{2} $
	83	$ 1 - 4.15e5iT - 2.71e13T^{2} $
	89	$ 1 + 2.04e6iT - 4.42e13T^{2} $
	97	$ 1 + 1.09e6iT - 8.07e13T^{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

−10.55106557234646084497744466216, −9.865302852778428016893556206146, −8.878758441016273170203119248385, −7.87358107581305892238182572710, −6.74594594810755381394232001736, −5.01926504424896846951808720293, −4.45789585787672918435148206619, −3.19533100369994420008028669067, −1.40528054018674921120832457116, −0.901690103486054421450845751483, 0.871277228110544982994515668925, 2.79334269628202235869531982837, 3.57565497296683822504921448749, 5.46747250657035413003223209217, 6.03940294262703626083750532250, 7.02568404485929809538437388466, 8.316037461288385540767041148870, 8.867042851299972856165972823947, 10.23395761856300687267369759269, 11.10902665407411634207208175630

Overview	Random
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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

\(L(4)\)	\(\approx\)	\(2.403930797\)
\(L(\frac12)\)	\(\approx\)	\(2.403930797\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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