LMFDB - L-function 2-600-5.4-c5-0-41

Dirichlet series

L(s) = 1

− 9i·3^-s − 21.5i·7^-s − 81·9^-s + 2.59·11^-s − 1.04e3i·13^-s − 1.39e3i·17^-s + 2.89e3·19^-s − 193.·21^-s + 3.60e3i·23^-s + 729i·27^-s − 3.12e3·29^-s + 1.37e3·31^-s − 23.3i·33^-s + 5.26e3i·37^-s − 9.42e3·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.166i·7^-s − 0.333·9^-s + 0.00646·11^-s − 1.71i·13^-s − 1.17i·17^-s + 1.83·19^-s − 0.0959·21^-s + 1.41i·23^-s + 0.192i·27^-s − 0.689·29^-s + 0.256·31^-s − 0.00373i·33^-s + 0.632i·37^-s − 0.991·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$600$ = $2^{3} \cdot 3 \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$96.2302$
Root analytic conductor:	$9.80970$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{600} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 600,\ (\ :5/2),\ -0.894 + 0.447i)$

Particular Values

$L(3)$	$\approx$	$1.554189848$
$L(\frac12)$	$\approx$	$1.554189848$
$L(\frac{7}{2})$		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 + 9iT $
	5	$ 1 $
good	7	$ 1 + 21.5iT - 1.68e4T^{2} $
	11	$ 1 - 2.59T + 1.61e5T^{2} $
	13	$ 1 + 1.04e3iT - 3.71e5T^{2} $
	17	$ 1 + 1.39e3iT - 1.41e6T^{2} $
	19	$ 1 - 2.89e3T + 2.47e6T^{2} $
	23	$ 1 - 3.60e3iT - 6.43e6T^{2} $
	29	$ 1 + 3.12e3T + 2.05e7T^{2} $
	31	$ 1 - 1.37e3T + 2.86e7T^{2} $
	37	$ 1 - 5.26e3iT - 6.93e7T^{2} $
	41	$ 1 + 1.71e3T + 1.15e8T^{2} $
	43	$ 1 + 1.34e4iT - 1.47e8T^{2} $
	47	$ 1 + 1.52e4iT - 2.29e8T^{2} $
	53	$ 1 + 9.44e3iT - 4.18e8T^{2} $
	59	$ 1 - 3.04e4T + 7.14e8T^{2} $
	61	$ 1 + 609.T + 8.44e8T^{2} $
	67	$ 1 + 1.46e4iT - 1.35e9T^{2} $
	71	$ 1 + 4.67e4T + 1.80e9T^{2} $
	73	$ 1 - 5.56e4iT - 2.07e9T^{2} $
	79	$ 1 + 5.95e4T + 3.07e9T^{2} $
	83	$ 1 + 5.18e4iT - 3.93e9T^{2} $
	89	$ 1 + 8.40e4T + 5.58e9T^{2} $
	97	$ 1 + 1.59e5iT - 8.58e9T^{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

−9.606822474212288364373203924037, −8.523225701487670358505830748296, −7.50742091750699149626299167328, −7.17189684275934688504369100873, −5.63062647409337713557727852658, −5.24998772682302161366652163443, −3.54850319461481710746136631990, −2.76318197430866101856744137782, −1.27896755493452956177428667056, −0.36530107218345886469498229508, 1.29875193847870088205715601470, 2.55465592281788465583904659232, 3.81674788986183055371796279020, 4.58205378048064226323037815401, 5.70070684216678029841466427220, 6.60605758770240497503792207877, 7.63954058495234045092784447437, 8.732901324839009802701475354504, 9.354832915666835122480986784518, 10.18858623158110617809061141011

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 $\F_{q}$
Belyi maps

\(L(3)\)	\(\approx\)	\(1.554189848\)
\(L(\frac12)\)	\(\approx\)	\(1.554189848\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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