LMFDB - L-function 2-75-5.4-c17-0-18

Dirichlet series

L(s) = 1

− 577. i·2^-s + 6.56e3i·3^-s − 2.02e5·4^-s + 3.79e6·6^-s + 2.79e7i·7^-s + 4.15e7i·8^-s − 4.30e7·9^-s + 6.58e8·11^-s − 1.33e9i·12^-s − 4.25e8i·13^-s + 1.61e10·14^-s − 2.59e9·16^-s − 4.12e10i·17^-s + 2.48e10i·18^-s − 1.11e11·19^-s + ⋯

L(s) = 1

− 1.59i·2^-s + 0.577i·3^-s − 1.54·4^-s + 0.921·6^-s + 1.83i·7^-s + 0.875i·8^-s − 0.333·9^-s + 0.925·11^-s − 0.893i·12^-s − 0.144i·13^-s + 2.92·14^-s − 0.151·16^-s − 1.43i·17^-s + 0.532i·18^-s − 1.50·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$75$ = $3 \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$137.416$
Root analytic conductor:	$11.7224$
Motivic weight:	$17$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{75} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 75,\ (\ :17/2),\ 0.894 + 0.447i)$

Particular Values

$L(9)$	$\approx$	$1.741966807$
$L(\frac12)$	$\approx$	$1.741966807$
$L(\frac{19}{2})$		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 - 6.56e3iT $
	5	$ 1 $
good	2	$ 1 + 577. iT - 1.31e5T^{2} $
	7	$ 1 - 2.79e7iT - 2.32e14T^{2} $
	11	$ 1 - 6.58e8T + 5.05e17T^{2} $
	13	$ 1 + 4.25e8iT - 8.65e18T^{2} $
	17	$ 1 + 4.12e10iT - 8.27e20T^{2} $
	19	$ 1 + 1.11e11T + 5.48e21T^{2} $
	23	$ 1 + 2.41e11iT - 1.41e23T^{2} $
	29	$ 1 - 9.10e11T + 7.25e24T^{2} $
	31	$ 1 - 6.61e12T + 2.25e25T^{2} $
	37	$ 1 + 2.53e13iT - 4.56e26T^{2} $
	41	$ 1 - 9.96e12T + 2.61e27T^{2} $
	43	$ 1 - 4.20e12iT - 5.87e27T^{2} $
	47	$ 1 + 1.54e14iT - 2.66e28T^{2} $
	53	$ 1 - 7.54e14iT - 2.05e29T^{2} $
	59	$ 1 - 6.28e14T + 1.27e30T^{2} $
	61	$ 1 - 3.95e14T + 2.24e30T^{2} $
	67	$ 1 - 6.34e15iT - 1.10e31T^{2} $
	71	$ 1 - 1.00e16T + 2.96e31T^{2} $
	73	$ 1 + 8.47e14iT - 4.74e31T^{2} $
	79	$ 1 + 4.80e15T + 1.81e32T^{2} $
	83	$ 1 - 6.98e15iT - 4.21e32T^{2} $
	89	$ 1 - 1.08e16T + 1.37e33T^{2} $
	97	$ 1 - 7.39e16iT - 5.95e33T^{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

−11.31366693094572262957840265256, −10.13492216786558690757990778038, −9.149290988870098307429665686477, −8.639989984492763149984715870217, −6.38914359424312875839922674705, −5.06258563638704429614454379536, −4.03967854164028207115099742068, −2.72027590045848428643365145417, −2.24497688601746412548691034430, −0.75084787068884372385662986881, 0.48898421514290813896889757727, 1.61574370280769191553131504428, 3.79236068392219734810657182216, 4.62839892318975316092748273701, 6.36952251519013069077370708918, 6.63713635146984575277885906610, 7.79696586209805442879583375818, 8.507064167513247617637617601396, 10.00948021599571991739315030198, 11.21363888111033513340833508394

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

\(L(9)\)	\(\approx\)	\(1.741966807\)
\(L(\frac12)\)	\(\approx\)	\(1.741966807\)
\(L(\frac{19}{2})\)		not available
\(L(1)\)		not available

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