LMFDB - L-function 2-8280-69.68-c1-0-78

Dirichlet series

L(s) = 1

− 5^-s − 3.49i·7^-s + 1.14·11^-s − 2.26·13^-s + 6.86·17^-s − 6.79i·19^-s + (−4.43 + 1.81i)23^-s + 25^-s + 2.09i·29^-s + 5.90·31^-s + 3.49i·35^-s + 2.76i·37^-s − 0.214i·41^-s − 9.72i·43^-s − 5.38i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s − 1.32i·7^-s + 0.346·11^-s − 0.629·13^-s + 1.66·17^-s − 1.55i·19^-s + (−0.925 + 0.378i)23^-s + 0.200·25^-s + 0.388i·29^-s + 1.06·31^-s + 0.591i·35^-s + 0.453i·37^-s − 0.0335i·41^-s − 1.48i·43^-s − 0.786i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$-0.537 + 0.843i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ -0.537 + 0.843i)$

Particular Values

$L(1)$	$\approx$	$1.518120856$
$L(\frac12)$	$\approx$	$1.518120856$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	23	$ 1 + (4.43 - 1.81i)T $
good	7	$ 1 + 3.49iT - 7T^{2} $
	11	$ 1 - 1.14T + 11T^{2} $
	13	$ 1 + 2.26T + 13T^{2} $
	17	$ 1 - 6.86T + 17T^{2} $
	19	$ 1 + 6.79iT - 19T^{2} $
	29	$ 1 - 2.09iT - 29T^{2} $
	31	$ 1 - 5.90T + 31T^{2} $
	37	$ 1 - 2.76iT - 37T^{2} $
	41	$ 1 + 0.214iT - 41T^{2} $
	43	$ 1 + 9.72iT - 43T^{2} $
	47	$ 1 + 5.38iT - 47T^{2} $
	53	$ 1 - 6.28T + 53T^{2} $
	59	$ 1 + 5.04iT - 59T^{2} $
	61	$ 1 - 2.18iT - 61T^{2} $
	67	$ 1 - 6.63iT - 67T^{2} $
	71	$ 1 + 2.63iT - 71T^{2} $
	73	$ 1 - 7.54T + 73T^{2} $
	79	$ 1 + 5.09iT - 79T^{2} $
	83	$ 1 - 1.54T + 83T^{2} $
	89	$ 1 - 0.845T + 89T^{2} $
	97	$ 1 + 5.15iT - 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

−7.41759499425540688543798898715, −7.11165695577450652303625632835, −6.35035983223815273376852969551, −5.33785647117956138040836092980, −4.74493219291532531331841029146, −3.91787979757782421995238552833, −3.41933400517730946868478997837, −2.42932132387721987619880686821, −1.18260581182477083578156269854, −0.41684094240266440632778056716, 1.11568399416791593508085224606, 2.15294927233548496184222558897, 2.93924760418822603728355535285, 3.72404037157303011315630969841, 4.51909052155094754006710908199, 5.41089159099486440211129765931, 5.92192864241179241206425678370, 6.49518964057357963326987865605, 7.66391998813577330642394480597, 7.940290710763313723389429683805

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.518120856\)
\(L(\frac12)\)	\(\approx\)	\(1.518120856\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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