LMFDB - L-function 2-825-5.4-c3-0-9

Dirichlet series

L(s) = 1

− 5.26i·2^-s + 3i·3^-s − 19.6·4^-s + 15.7·6^-s − 10.3i·7^-s + 61.4i·8^-s − 9·9^-s + 11·11^-s − 59.0i·12^-s − 63.9i·13^-s − 54.3·14^-s + 165.·16^-s + 17.1i·17^-s + 47.3i·18^-s − 90.2·19^-s + ⋯

L(s) = 1

− 1.86i·2^-s + 0.577i·3^-s − 2.46·4^-s + 1.07·6^-s − 0.557i·7^-s + 2.71i·8^-s − 0.333·9^-s + 0.301·11^-s − 1.42i·12^-s − 1.36i·13^-s − 1.03·14^-s + 2.59·16^-s + 0.244i·17^-s + 0.620i·18^-s − 1.08·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$825$ = $3 \cdot 5^{2} \cdot 11$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$48.6765$
Root analytic conductor:	$6.97686$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{825} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$0.8683758708$
$L(\frac12)$	$\approx$	$0.8683758708$
$L(\frac{5}{2})$		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 - 3iT $
	5	$ 1 $
	11	$ 1 - 11T $
good	2	$ 1 + 5.26iT - 8T^{2} $
	7	$ 1 + 10.3iT - 343T^{2} $
	13	$ 1 + 63.9iT - 2.19e3T^{2} $
	17	$ 1 - 17.1iT - 4.91e3T^{2} $
	19	$ 1 + 90.2T + 6.85e3T^{2} $
	23	$ 1 - 212. iT - 1.21e4T^{2} $
	29	$ 1 + 57.5T + 2.43e4T^{2} $
	31	$ 1 + 141.T + 2.97e4T^{2} $
	37	$ 1 + 257. iT - 5.06e4T^{2} $
	41	$ 1 + 225.T + 6.89e4T^{2} $
	43	$ 1 - 347. iT - 7.95e4T^{2} $
	47	$ 1 - 404. iT - 1.03e5T^{2} $
	53	$ 1 + 259. iT - 1.48e5T^{2} $
	59	$ 1 - 853.T + 2.05e5T^{2} $
	61	$ 1 + 203.T + 2.26e5T^{2} $
	67	$ 1 - 266. iT - 3.00e5T^{2} $
	71	$ 1 - 92.4T + 3.57e5T^{2} $
	73	$ 1 - 242. iT - 3.89e5T^{2} $
	79	$ 1 - 1.02e3T + 4.93e5T^{2} $
	83	$ 1 + 706. iT - 5.71e5T^{2} $
	89	$ 1 - 440.T + 7.04e5T^{2} $
	97	$ 1 + 197. iT - 9.12e5T^{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

−9.987803004276853830135248231135, −9.353832691099224331669861002334, −8.487161080627044418988778611796, −7.53780469616480412008211462934, −5.82232252509367989238130665924, −4.93211562044559100897948968175, −3.85323246458169940572605926418, −3.39635407773799364715554303795, −2.13871012016743284835804207660, −0.920839449375790964590036914452, 0.29333210226067697247864627490, 2.10591576118162003286585364675, 3.94590424797836244282359280031, 4.83794198636038342370295609915, 5.80093538313168097969524945206, 6.78557383137548751945948070641, 6.85270693995958260047025808178, 8.204613395578948146656365236933, 8.707827551874752072183790133762, 9.317123083981973370888742840184

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8683758708\)
\(L(\frac12)\)	\(\approx\)	\(0.8683758708\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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