LMFDB - L-function 2-4100-5.4-c1-0-56

Dirichlet series

L(s) = 1

− 0.687i·3^-s − 4.33i·7^-s + 2.52·9^-s + 0.296·11^-s − 4.32i·13^-s + 1.86i·17^-s − 6.48·19^-s − 2.98·21^-s − 3.04i·23^-s − 3.80i·27^-s + 7.67·29^-s + 2.34·31^-s − 0.204i·33^-s − 9.25i·37^-s − 2.97·39^-s + ⋯

L(s) = 1

− 0.397i·3^-s − 1.63i·7^-s + 0.842·9^-s + 0.0894·11^-s − 1.20i·13^-s + 0.453i·17^-s − 1.48·19^-s − 0.651·21^-s − 0.634i·23^-s − 0.731i·27^-s + 1.42·29^-s + 0.421·31^-s − 0.0355i·33^-s − 1.52i·37^-s − 0.476·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4100$ = $2^{2} \cdot 5^{2} \cdot 41$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$32.7386$
Root analytic conductor:	$5.72177$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4100} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4100,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.584436463$
$L(\frac12)$	$\approx$	$1.584436463$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	41	$ 1 + T $
good	3	$ 1 + 0.687iT - 3T^{2} $
	7	$ 1 + 4.33iT - 7T^{2} $
	11	$ 1 - 0.296T + 11T^{2} $
	13	$ 1 + 4.32iT - 13T^{2} $
	17	$ 1 - 1.86iT - 17T^{2} $
	19	$ 1 + 6.48T + 19T^{2} $
	23	$ 1 + 3.04iT - 23T^{2} $
	29	$ 1 - 7.67T + 29T^{2} $
	31	$ 1 - 2.34T + 31T^{2} $
	37	$ 1 + 9.25iT - 37T^{2} $
	43	$ 1 - 12.5iT - 43T^{2} $
	47	$ 1 + 5.21iT - 47T^{2} $
	53	$ 1 + 13.7iT - 53T^{2} $
	59	$ 1 + 8.96T + 59T^{2} $
	61	$ 1 - 2.45T + 61T^{2} $
	67	$ 1 + 0.634iT - 67T^{2} $
	71	$ 1 + 6.73T + 71T^{2} $
	73	$ 1 - 11.8iT - 73T^{2} $
	79	$ 1 + 7.73T + 79T^{2} $
	83	$ 1 - 16.0iT - 83T^{2} $
	89	$ 1 - 17.2T + 89T^{2} $
	97	$ 1 + 9.31iT - 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

−8.028068105930856275030588527715, −7.40529395205109403627221258241, −6.64596453065056747439456346119, −6.22771436098814106193743170633, −4.92261674228948326785265933317, −4.27041710311895303359146569868, −3.63523353766872064922824416722, −2.47089662539777241991551712332, −1.30855130303952740197716605288, −0.46580072128457848446617534854, 1.55153707777963086815243898552, 2.35647898542012910712020735450, 3.27960113365835371693459345338, 4.50634912663032561647396982490, 4.72129335345069036955656452436, 5.89225695427294057908184143443, 6.43151241037663733698886047094, 7.18799721587748756560164264167, 8.191633152004627562749249823585, 8.986717728766352446267445252816

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.584436463\)
\(L(\frac12)\)	\(\approx\)	\(1.584436463\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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