LMFDB - L-function 2-6336-12.11-c1-0-52

Dirichlet series

L(s) = 1

− 4.08i·5^-s + 4.08i·7^-s + 11^-s + 5.74·13^-s − 1.62i·17^-s − 6.07i·19^-s + 6.92·23^-s − 11.6·25^-s + 3.46i·29^-s − 1.83i·31^-s + 16.6·35^-s − 5.47·37^-s − 1.83i·41^-s + 5.75i·43^-s + 7.21·47^-s + ⋯

L(s) = 1

− 1.82i·5^-s + 1.54i·7^-s + 0.301·11^-s + 1.59·13^-s − 0.395i·17^-s − 1.39i·19^-s + 1.44·23^-s − 2.33·25^-s + 0.643i·29^-s − 0.329i·31^-s + 2.81·35^-s − 0.899·37^-s − 0.286i·41^-s + 0.878i·43^-s + 1.05·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6336$ = $2^{6} \cdot 3^{2} \cdot 11$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$50.5932$
Root analytic conductor:	$7.11289$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6336} (3455, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6336,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$2.281212695$
$L(\frac12)$	$\approx$	$2.281212695$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	11	$ 1 - T $
good	5	$ 1 + 4.08iT - 5T^{2} $
	7	$ 1 - 4.08iT - 7T^{2} $
	13	$ 1 - 5.74T + 13T^{2} $
	17	$ 1 + 1.62iT - 17T^{2} $
	19	$ 1 + 6.07iT - 19T^{2} $
	23	$ 1 - 6.92T + 23T^{2} $
	29	$ 1 - 3.46iT - 29T^{2} $
	31	$ 1 + 1.83iT - 31T^{2} $
	37	$ 1 + 5.47T + 37T^{2} $
	41	$ 1 + 1.83iT - 41T^{2} $
	43	$ 1 - 5.75iT - 43T^{2} $
	47	$ 1 - 7.21T + 47T^{2} $
	53	$ 1 - 11.0iT - 53T^{2} $
	59	$ 1 - 6.65T + 59T^{2} $
	61	$ 1 - 1.74T + 61T^{2} $
	67	$ 1 + 11.9iT - 67T^{2} $
	71	$ 1 - 6.92T + 71T^{2} $
	73	$ 1 - 4.65T + 73T^{2} $
	79	$ 1 - 0.825iT - 79T^{2} $
	83	$ 1 + 4.67T + 83T^{2} $
	89	$ 1 + 3.00iT - 89T^{2} $
	97	$ 1 + 12.6T + 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

−8.343946015797534811023020554818, −7.25413370975604603076667512243, −6.32472904678980193977635364431, −5.63917653585219476480816695060, −5.10297335449293302254679158768, −4.53694012950601386070719645898, −3.51880677428042463746858939016, −2.55388101684956775118197880807, −1.51027461967043496225278876161, −0.73655421591423925575656701931, 0.958396921994699882290041829666, 1.96119921551081672734610617960, 3.19511122653511172443350006756, 3.70594465696447321106635975738, 4.08785853442779648809754138785, 5.49970730813618361054576111461, 6.23674093492483725941042430771, 6.88390259180888138735936244884, 7.16458110453779846229311553914, 8.034080091455306814973096735266

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

\(L(1)\)	\(\approx\)	\(2.281212695\)
\(L(\frac12)\)	\(\approx\)	\(2.281212695\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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