LMFDB - L-function 2-2450-5.4-c1-0-31

Dirichlet series

L(s) = 1

− i·2^-s + 3.41i·3^-s − 4^-s + 3.41·6^-s + i·8^-s − 8.65·9^-s − 0.828·11^-s − 3.41i·12^-s − 4.82i·13^-s + 16^-s − 2.58i·17^-s + 8.65i·18^-s + 0.585·19^-s + 0.828i·22^-s + 1.17i·23^-s − 3.41·24^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 1.97i·3^-s − 0.5·4^-s + 1.39·6^-s + 0.353i·8^-s − 2.88·9^-s − 0.249·11^-s − 0.985i·12^-s − 1.33i·13^-s + 0.250·16^-s − 0.627i·17^-s + 2.04i·18^-s + 0.134·19^-s + 0.176i·22^-s + 0.244i·23^-s − 0.696·24^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2450$ = $2 \cdot 5^{2} \cdot 7^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$19.5633$
Root analytic conductor:	$4.42304$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2450} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2450,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.177946594$
$L(\frac12)$	$\approx$	$1.177946594$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 $
	7	$ 1 $
good	3	$ 1 - 3.41iT - 3T^{2} $
	11	$ 1 + 0.828T + 11T^{2} $
	13	$ 1 + 4.82iT - 13T^{2} $
	17	$ 1 + 2.58iT - 17T^{2} $
	19	$ 1 - 0.585T + 19T^{2} $
	23	$ 1 - 1.17iT - 23T^{2} $
	29	$ 1 - 4.82T + 29T^{2} $
	31	$ 1 - 2.82T + 31T^{2} $
	37	$ 1 + 7.65iT - 37T^{2} $
	41	$ 1 - 3.07T + 41T^{2} $
	43	$ 1 - 8.82iT - 43T^{2} $
	47	$ 1 + 5.17iT - 47T^{2} $
	53	$ 1 + 6.48iT - 53T^{2} $
	59	$ 1 - 8.58T + 59T^{2} $
	61	$ 1 + 9.31T + 61T^{2} $
	67	$ 1 - 1.65iT - 67T^{2} $
	71	$ 1 + 4.48T + 71T^{2} $
	73	$ 1 + 9.41iT - 73T^{2} $
	79	$ 1 - 6.82T + 79T^{2} $
	83	$ 1 + 2.24iT - 83T^{2} $
	89	$ 1 - 12.7T + 89T^{2} $
	97	$ 1 - 7.75iT - 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

−9.196552723018125136410580524337, −8.416068024100234669932902608634, −7.73947502235944497214944453403, −6.16878573231872667256803188399, −5.32774007321507042114450348124, −4.85903324126666250537723549933, −3.97137077215654833861064425118, −3.18008836411010787699930619470, −2.58272739885260420126331420069, −0.47487567633079684544706117702, 1.00935103571065460789243477596, 2.00489898009576895277026705974, 3.00101854427892507474109576660, 4.35679095534201543586701553313, 5.43867255982711025489240370993, 6.26652502480383816346371591988, 6.69229598301530869098820450411, 7.34524187923658713334585302426, 8.110009255302859125111659706007, 8.600206130795989397522428422604

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.177946594\)
\(L(\frac12)\)	\(\approx\)	\(1.177946594\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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