LMFDB - L-function 2-4004-13.12-c1-0-38

Dirichlet series

L(s) = 1

− 2.41·3^-s + 3.42i·5^-s − i·7^-s + 2.80·9^-s − i·11^-s + (3.35 + 1.32i)13^-s − 8.26i·15^-s − 0.705·17^-s + 0.998i·19^-s + 2.41i·21^-s + 2.09·23^-s − 6.74·25^-s + 0.459·27^-s + 1.73·29^-s − 8.34i·31^-s + ⋯

L(s) = 1

− 1.39·3^-s + 1.53i·5^-s − 0.377i·7^-s + 0.936·9^-s − 0.301i·11^-s + (0.929 + 0.367i)13^-s − 2.13i·15^-s − 0.171·17^-s + 0.229i·19^-s + 0.525i·21^-s + 0.436·23^-s − 1.34·25^-s + 0.0885·27^-s + 0.322·29^-s − 1.49i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4004$ = $2^{2} \cdot 7 \cdot 11 \cdot 13$
Sign:	$0.929 + 0.367i$
Analytic conductor:	$31.9721$
Root analytic conductor:	$5.65438$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4004} (2157, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4004,\ (\ :1/2),\ 0.929 + 0.367i)$

Particular Values

$L(1)$	$\approx$	$0.8724327791$
$L(\frac12)$	$\approx$	$0.8724327791$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	7	$ 1 + iT $
	11	$ 1 + iT $
	13	$ 1 + (-3.35 - 1.32i)T $
good	3	$ 1 + 2.41T + 3T^{2} $
	5	$ 1 - 3.42iT - 5T^{2} $
	17	$ 1 + 0.705T + 17T^{2} $
	19	$ 1 - 0.998iT - 19T^{2} $
	23	$ 1 - 2.09T + 23T^{2} $
	29	$ 1 - 1.73T + 29T^{2} $
	31	$ 1 + 8.34iT - 31T^{2} $
	37	$ 1 - 6.44iT - 37T^{2} $
	41	$ 1 + 8.10iT - 41T^{2} $
	43	$ 1 + 3.25T + 43T^{2} $
	47	$ 1 + 10.5iT - 47T^{2} $
	53	$ 1 + 5.72T + 53T^{2} $
	59	$ 1 + 3.36iT - 59T^{2} $
	61	$ 1 + 9.47T + 61T^{2} $
	67	$ 1 - 1.90iT - 67T^{2} $
	71	$ 1 - 0.177iT - 71T^{2} $
	73	$ 1 + 13.0iT - 73T^{2} $
	79	$ 1 + 1.06T + 79T^{2} $
	83	$ 1 + 4.13iT - 83T^{2} $
	89	$ 1 + 1.28iT - 89T^{2} $
	97	$ 1 - 1.32iT - 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

−8.262373435568238010337897629581, −7.39043797920464217169374877868, −6.70015722658503732634488297660, −6.28933366561543228744365614446, −5.68538755116835407263465603759, −4.70384876846317206268596135408, −3.78595724195082969442954668233, −3.03397823018492877103110824922, −1.79673959433729159449855200968, −0.42355178519362536503489342662, 0.851523446026307068739867676255, 1.53244926208308059182612384535, 3.07252561589344296658423348183, 4.35384832165693075671013900946, 4.84804139794302062388589237840, 5.47707802528784813423025865850, 6.08601759725434910354135766242, 6.79607088178518516332205343103, 7.84687634057449669225924872057, 8.600352991857556544441413587491

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8724327791\)
\(L(\frac12)\)	\(\approx\)	\(0.8724327791\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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