LMFDB - L-function 2-2300-5.4-c1-0-11

Dirichlet series

L(s) = 1

− 0.474i·3^-s + 4.21i·7^-s + 2.77·9^-s + 3.26·11^-s − 0.439i·13^-s + 2.94i·17^-s + 1.26·19^-s + 2·21^-s + i·23^-s − 2.74i·27^-s − 5.51·29^-s − 0.249·31^-s − 1.54i·33^-s + 8.49i·37^-s − 0.208·39^-s + ⋯

L(s) = 1

− 0.273i·3^-s + 1.59i·7^-s + 0.924·9^-s + 0.984·11^-s − 0.122i·13^-s + 0.715i·17^-s + 0.290·19^-s + 0.436·21^-s + 0.208i·23^-s − 0.527i·27^-s − 1.02·29^-s − 0.0447·31^-s − 0.269i·33^-s + 1.39i·37^-s − 0.0334·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$18.3655$
Root analytic conductor:	$4.28550$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1749, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :1/2),\ 0.447 - 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.939581133$
$L(\frac12)$	$\approx$	$1.939581133$
$L(\frac{3}{2})$		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 $
	23	$ 1 - iT $
good	3	$ 1 + 0.474iT - 3T^{2} $
	7	$ 1 - 4.21iT - 7T^{2} $
	11	$ 1 - 3.26T + 11T^{2} $
	13	$ 1 + 0.439iT - 13T^{2} $
	17	$ 1 - 2.94iT - 17T^{2} $
	19	$ 1 - 1.26T + 19T^{2} $
	29	$ 1 + 5.51T + 29T^{2} $
	31	$ 1 + 0.249T + 31T^{2} $
	37	$ 1 - 8.49iT - 37T^{2} $
	41	$ 1 - 2.86T + 41T^{2} $
	43	$ 1 + 1.26iT - 43T^{2} $
	47	$ 1 + 7.25iT - 47T^{2} $
	53	$ 1 - 9.48iT - 53T^{2} $
	59	$ 1 - 0.265T + 59T^{2} $
	61	$ 1 + 1.48T + 61T^{2} $
	67	$ 1 + 3.54iT - 67T^{2} $
	71	$ 1 - 6.82T + 71T^{2} $
	73	$ 1 - 1.08iT - 73T^{2} $
	79	$ 1 + 11.1T + 79T^{2} $
	83	$ 1 + 0.316iT - 83T^{2} $
	89	$ 1 + 8.36T + 89T^{2} $
	97	$ 1 - 9.41iT - 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

−9.142684039942170953822470498730, −8.452068713499586879113865356093, −7.63146722246298895706115183674, −6.72800389531934770126690237050, −6.06013849136184688469225921369, −5.31824591929813819682152149416, −4.30235287724818354090014812223, −3.36497945770705556017326292565, −2.20567850030517188627788734552, −1.37143753572090144994714685428, 0.72819592688068853549077723567, 1.75432982296671438791104751616, 3.33656660592408159811013639223, 4.10526140575241063146741321666, 4.55965120102629942000744637432, 5.72679573620893157201110917916, 6.88084162266859794283330142230, 7.15890896147574226847186867204, 7.925149040166066302152731270882, 9.115698558716172452015529753108

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.939581133\)
\(L(\frac12)\)	\(\approx\)	\(1.939581133\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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