LMFDB - L-function 2-2300-115.114-c2-0-71

Dirichlet series

L(s) = 1

− 5.43i·3^-s − 12.3·7^-s − 20.5·9^-s + 0.283i·11^-s − 18.2i·13^-s − 24.7·17^-s − 5.66i·19^-s + 67.0i·21^-s + (−22.2 − 5.92i)23^-s + 62.6i·27^-s + 36.9·29^-s − 13.8·31^-s + 1.53·33^-s − 48.0·37^-s − 98.9·39^-s + ⋯

L(s) = 1

− 1.81i·3^-s − 1.76·7^-s − 2.28·9^-s + 0.0257i·11^-s − 1.40i·13^-s − 1.45·17^-s − 0.298i·19^-s + 3.19i·21^-s + (−0.966 − 0.257i)23^-s + 2.31i·27^-s + 1.27·29^-s − 0.445·31^-s + 0.0466·33^-s − 1.29·37^-s − 2.53·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$0.979 - 0.201i$
Analytic conductor:	$62.6704$
Root analytic conductor:	$7.91646$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :1),\ 0.979 - 0.201i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.02280862576$
$L(\frac12)$	$\approx$	$0.02280862576$
$L(2)$		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 $
	23	$ 1 + (22.2 + 5.92i)T $
good	3	$ 1 + 5.43iT - 9T^{2} $
	7	$ 1 + 12.3T + 49T^{2} $
	11	$ 1 - 0.283iT - 121T^{2} $
	13	$ 1 + 18.2iT - 169T^{2} $
	17	$ 1 + 24.7T + 289T^{2} $
	19	$ 1 + 5.66iT - 361T^{2} $
	29	$ 1 - 36.9T + 841T^{2} $
	31	$ 1 + 13.8T + 961T^{2} $
	37	$ 1 + 48.0T + 1.36e3T^{2} $
	41	$ 1 + 30.8T + 1.68e3T^{2} $
	43	$ 1 + 10.2T + 1.84e3T^{2} $
	47	$ 1 + 16.8iT - 2.20e3T^{2} $
	53	$ 1 - 20.2T + 2.80e3T^{2} $
	59	$ 1 - 12.1T + 3.48e3T^{2} $
	61	$ 1 + 109. iT - 3.72e3T^{2} $
	67	$ 1 - 46.9T + 4.48e3T^{2} $
	71	$ 1 - 110.T + 5.04e3T^{2} $
	73	$ 1 + 102. iT - 5.32e3T^{2} $
	79	$ 1 - 88.9iT - 6.24e3T^{2} $
	83	$ 1 + 58.8T + 6.88e3T^{2} $
	89	$ 1 - 150. iT - 7.92e3T^{2} $
	97	$ 1 + 37.2T + 9.40e3T^{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

−8.110039079771015615897284489329, −7.00210909262059175002939388385, −6.68858040042736890581264840829, −6.08529377622399833719628348096, −5.20303888537367468206024579345, −3.61326689715170359147283070081, −2.82196343501695421249942722356, −2.05748754235973453299961821724, −0.60211457670457575793952316926, −0.008894326565273641604386648832, 2.28885612055429573180730605103, 3.29066347698681425170473673143, 3.96896335865103615890425527934, 4.54341423175129011730435896945, 5.59374398233147268100445093164, 6.41133104025600428543212381814, 6.96865939390545043201235868155, 8.587010208792664507586362378002, 8.922344658754565384225385623774, 9.697655686051125868934047507397

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.02280862576\)
\(L(\frac12)\)	\(\approx\)	\(0.02280862576\)
\(L(2)\)		not available
\(L(1)\)		not available

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