LMFDB - L-function 2-1035-69.68-c1-0-15

Dirichlet series

L(s) = 1

− 0.122i·2^-s + 1.98·4^-s + 5^-s + 2.75i·7^-s − 0.487i·8^-s − 0.122i·10^-s + 2.62·11^-s − 6.63·13^-s + 0.336·14^-s + 3.91·16^-s + 6.75·17^-s + 7.08i·19^-s + 1.98·20^-s − 0.321i·22^-s + (0.372 − 4.78i)23^-s + ⋯

L(s) = 1

− 0.0864i·2^-s + 0.992·4^-s + 0.447·5^-s + 1.04i·7^-s − 0.172i·8^-s − 0.0386i·10^-s + 0.792·11^-s − 1.83·13^-s + 0.0900·14^-s + 0.977·16^-s + 1.63·17^-s + 1.62i·19^-s + 0.443·20^-s − 0.0685i·22^-s + (0.0775 − 0.996i)23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1035$ = $3^{2} \cdot 5 \cdot 23$
Sign:	$0.858 - 0.512i$
Analytic conductor:	$8.26451$
Root analytic conductor:	$2.87480$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1035} (206, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1035,\ (\ :1/2),\ 0.858 - 0.512i)$

Particular Values

$L(1)$	$\approx$	$2.229577638$
$L(\frac12)$	$\approx$	$2.229577638$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 - T $
	23	$ 1 + (-0.372 + 4.78i)T $
good	2	$ 1 + 0.122iT - 2T^{2} $
	7	$ 1 - 2.75iT - 7T^{2} $
	11	$ 1 - 2.62T + 11T^{2} $
	13	$ 1 + 6.63T + 13T^{2} $
	17	$ 1 - 6.75T + 17T^{2} $
	19	$ 1 - 7.08iT - 19T^{2} $
	29	$ 1 - 4.57iT - 29T^{2} $
	31	$ 1 - 0.118T + 31T^{2} $
	37	$ 1 + 5.15iT - 37T^{2} $
	41	$ 1 - 8.28iT - 41T^{2} $
	43	$ 1 - 0.204iT - 43T^{2} $
	47	$ 1 + 5.93iT - 47T^{2} $
	53	$ 1 + 9.26T + 53T^{2} $
	59	$ 1 + 7.15iT - 59T^{2} $
	61	$ 1 - 7.55iT - 61T^{2} $
	67	$ 1 + 7.69iT - 67T^{2} $
	71	$ 1 + 9.43iT - 71T^{2} $
	73	$ 1 - 3.84T + 73T^{2} $
	79	$ 1 - 7.34iT - 79T^{2} $
	83	$ 1 - 8.04T + 83T^{2} $
	89	$ 1 - 2.97T + 89T^{2} $
	97	$ 1 - 10.1iT - 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

−9.925587513672739424437647113400, −9.443343744474722561241399534861, −8.225490620871515276671120481982, −7.50085427132987562909353648738, −6.53473220088601760811361265552, −5.79514364294128280007695541928, −5.02371462736782673766112776726, −3.46045549002366227438830415195, −2.51534338695833129953337829177, −1.57718182004118538205545817186, 1.09265152637452057990405878600, 2.41067231194697872097572042434, 3.41582697156469844928046290908, 4.69904880143478071973042861285, 5.62140631360989595728436506509, 6.69296569974149490997162800994, 7.30491964640361442994157719847, 7.82212799787322349068311600530, 9.312606078274411552097091155417, 9.899526873733981126423992428883

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.229577638\)
\(L(\frac12)\)	\(\approx\)	\(2.229577638\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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