LMFDB - L-function 2-3100-155.154-c2-0-11

Dirichlet series

L(s) = 1

+ 4.52·3^-s − 3.21i·7^-s + 11.4·9^-s + 8.05i·11^-s − 15.5·13^-s − 25.5·17^-s + 15.6·19^-s − 14.5i·21^-s − 23.5·23^-s + 11.0·27^-s + 38.7i·29^-s + (4.56 − 30.6i)31^-s + 36.4i·33^-s − 2.94·37^-s − 70.1·39^-s + ⋯

L(s) = 1

+ 1.50·3^-s − 0.459i·7^-s + 1.27·9^-s + 0.732i·11^-s − 1.19·13^-s − 1.50·17^-s + 0.823·19^-s − 0.692i·21^-s − 1.02·23^-s + 0.407·27^-s + 1.33i·29^-s + (0.147 − 0.989i)31^-s + 1.10i·33^-s − 0.0795·37^-s − 1.79·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$-0.818 - 0.574i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ -0.818 - 0.574i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.026311083$
$L(\frac12)$	$\approx$	$1.026311083$
$L(2)$		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 $
	31	$ 1 + (-4.56 + 30.6i)T $
good	3	$ 1 - 4.52T + 9T^{2} $
	7	$ 1 + 3.21iT - 49T^{2} $
	11	$ 1 - 8.05iT - 121T^{2} $
	13	$ 1 + 15.5T + 169T^{2} $
	17	$ 1 + 25.5T + 289T^{2} $
	19	$ 1 - 15.6T + 361T^{2} $
	23	$ 1 + 23.5T + 529T^{2} $
	29	$ 1 - 38.7iT - 841T^{2} $
	37	$ 1 + 2.94T + 1.36e3T^{2} $
	41	$ 1 - 6.51T + 1.68e3T^{2} $
	43	$ 1 - 4.52T + 1.84e3T^{2} $
	47	$ 1 - 72.1iT - 2.20e3T^{2} $
	53	$ 1 + 71.3T + 2.80e3T^{2} $
	59	$ 1 + 42.9T + 3.48e3T^{2} $
	61	$ 1 - 83.9iT - 3.72e3T^{2} $
	67	$ 1 - 82.8iT - 4.48e3T^{2} $
	71	$ 1 + 74.2T + 5.04e3T^{2} $
	73	$ 1 + 5.11T + 5.32e3T^{2} $
	79	$ 1 - 8.23iT - 6.24e3T^{2} $
	83	$ 1 + 103.T + 6.88e3T^{2} $
	89	$ 1 + 114. iT - 7.92e3T^{2} $
	97	$ 1 + 21.8iT - 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

−8.880232829371504826962324036541, −8.046584376599608781220046134831, −7.36608142058169775987810541910, −7.01675226698415279153826603569, −5.80875705631834345588977702941, −4.55856435399276659969670921029, −4.21662419143456278464776760591, −3.05480961293234939631424469450, −2.41480415500772770973059641877, −1.54941672313410226361432843040, 0.15555795276572203446673936831, 1.85716731491148182830419221881, 2.50792035515553413323711963596, 3.24309819991785164406510461435, 4.14531535887297734050645918766, 4.99412867293545320469474202050, 6.01399191059397637386720104174, 6.90140402860090050327741567699, 7.71180907857407048182537883054, 8.271225231784711673110998592168

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.026311083\)
\(L(\frac12)\)	\(\approx\)	\(1.026311083\)
\(L(2)\)		not available
\(L(1)\)		not available

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