LMFDB - L-function 2-342-19.18-c4-0-4

Dirichlet series

L(s) = 1

− 2.82i·2^-s − 8.00·4^-s − 41.6·5^-s − 62.4·7^-s + 22.6i·8^-s + 117. i·10^-s − 122.·11^-s − 68.9i·13^-s + 176. i·14^-s + 64.0·16^-s − 297.·17^-s + (−195. + 303. i)19^-s + 332.·20^-s + 347. i·22^-s − 268.·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.500·4^-s − 1.66·5^-s − 1.27·7^-s + 0.353i·8^-s + 1.17i·10^-s − 1.01·11^-s − 0.408i·13^-s + 0.900i·14^-s + 0.250·16^-s − 1.02·17^-s + (−0.541 + 0.840i)19^-s + 0.832·20^-s + 0.718i·22^-s − 0.507·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$342$ = $2 \cdot 3^{2} \cdot 19$
Sign:	$0.840 + 0.541i$
Analytic conductor:	$35.3525$
Root analytic conductor:	$5.94579$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{342} (37, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 342,\ (\ :2),\ 0.840 + 0.541i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.2636931360$
$L(\frac12)$	$\approx$	$0.2636931360$
$L(3)$		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 + 2.82iT $
	3	$ 1 $
	19	$ 1 + (195. - 303. i)T $
good	5	$ 1 + 41.6T + 625T^{2} $
	7	$ 1 + 62.4T + 2.40e3T^{2} $
	11	$ 1 + 122.T + 1.46e4T^{2} $
	13	$ 1 + 68.9iT - 2.85e4T^{2} $
	17	$ 1 + 297.T + 8.35e4T^{2} $
	23	$ 1 + 268.T + 2.79e5T^{2} $
	29	$ 1 + 561. iT - 7.07e5T^{2} $
	31	$ 1 - 252. iT - 9.23e5T^{2} $
	37	$ 1 + 2.40e3iT - 1.87e6T^{2} $
	41	$ 1 + 690. iT - 2.82e6T^{2} $
	43	$ 1 - 218.T + 3.41e6T^{2} $
	47	$ 1 - 83.1T + 4.87e6T^{2} $
	53	$ 1 - 4.38e3iT - 7.89e6T^{2} $
	59	$ 1 + 476. iT - 1.21e7T^{2} $
	61	$ 1 - 3.96e3T + 1.38e7T^{2} $
	67	$ 1 + 5.11e3iT - 2.01e7T^{2} $
	71	$ 1 - 8.18e3iT - 2.54e7T^{2} $
	73	$ 1 + 7.34e3T + 2.83e7T^{2} $
	79	$ 1 - 1.48e3iT - 3.89e7T^{2} $
	83	$ 1 + 5.34e3T + 4.74e7T^{2} $
	89	$ 1 + 1.11e4iT - 6.27e7T^{2} $
	97	$ 1 + 1.07e3iT - 8.85e7T^{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

−10.82205404124829622272463369161, −10.14370164992412291746353166861, −8.940533683587277106228262355941, −8.054908398123036412868849516385, −7.18917557313868158652253751916, −5.84943031068784135570538889150, −4.35446486310095731372967005768, −3.60321735081259257813228492503, −2.53812175616650603826502296734, −0.34128637855499671756155367090, 0.24438477800554772552297965093, 2.88839049028144617443152947074, 3.98671778366609736595454615714, 4.91784538518384414434834555489, 6.46278318948177679609199451571, 7.07889228103973116824916601280, 8.099383000266909532544385584028, 8.823571154983037405226889400850, 9.968227305783616843920709859569, 11.02468583223399150879832605283

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 $\F_{q}$

\(L(\frac{5}{2})\)	\(\approx\)	\(0.2636931360\)
\(L(\frac12)\)	\(\approx\)	\(0.2636931360\)
\(L(3)\)		not available
\(L(1)\)		not available

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