LMFDB - L-function 2-3675-15.14-c0-0-1

Dirichlet series

L(s) = 1

− 2^-s + (0.707 + 0.707i)3^-s + (−0.707 − 0.707i)6^-s + 8^-s + 1.00i·9^-s + i·11^-s + 1.41i·13^-s − 16^-s − 1.41·17^-s − 1.00i·18^-s − i·22^-s − 23^-s + (0.707 + 0.707i)24^-s − 1.41i·26^-s + (−0.707 + 0.707i)27^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3675$ = $3 \cdot 5^{2} \cdot 7^{2}$
Sign:	$-0.948 - 0.316i$
Analytic conductor:	$1.83406$
Root analytic conductor:	$1.35427$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3675} (2549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3675,\ (\ :0),\ -0.948 - 0.316i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5568720140$
$L(\frac12)$	$\approx$	$0.5568720140$
$L(1)$		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 + (-0.707 - 0.707i)T $
	5	$ 1 $
	7	$ 1 $
good	2	$ 1 + T + T^{2} $
	11	$ 1 - iT - T^{2} $
	13	$ 1 - 1.41iT - T^{2} $
	17	$ 1 + 1.41T + T^{2} $
	19	$ 1 + T^{2} $
	23	$ 1 + T + T^{2} $
	29	$ 1 + iT - T^{2} $
	31	$ 1 + T^{2} $
	37	$ 1 - iT - T^{2} $
	41	$ 1 + 1.41iT - T^{2} $
	43	$ 1 - iT - T^{2} $
	47	$ 1 + T^{2} $
	53	$ 1 + T^{2} $
	59	$ 1 + 1.41iT - T^{2} $
	61	$ 1 + 1.41T + T^{2} $
	67	$ 1 + iT - T^{2} $
	71	$ 1 - iT - T^{2} $
	73	$ 1 - T^{2} $
	79	$ 1 - T + T^{2} $
	83	$ 1 + T^{2} $
	89	$ 1 - 1.41iT - T^{2} $
	97	$ 1 - T^{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

−9.192946615647820360023325919207, −8.447673436058734168069123662995, −7.85081341484344483932483860363, −7.08706979724893767241091599894, −6.33197735699405715670802193157, −4.88715909145841024009092962426, −4.44420968322169307392502041715, −3.81894230768888089259797642934, −2.29190503396621621049070643934, −1.81608384955572879865044446536, 0.40656867756388143796712169822, 1.54175561967873608513316911049, 2.59382639009614891449122552963, 3.49112969184769902233810546635, 4.43301633134307887712851119469, 5.58065471804084369965885004206, 6.34854059475098032613612417502, 7.26013181267573919130345336325, 7.83307958526536724639068522199, 8.485925429962475856757411600388

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

\(L(\frac{1}{2})\)	\(\approx\)	\(0.5568720140\)
\(L(\frac12)\)	\(\approx\)	\(0.5568720140\)
\(L(1)\)		not available
\(L(1)\)		not available

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