LMFDB - L-function 2-3100-31.30-c2-0-94

Dirichlet series

L(s) = 1

− 4.11i·3^-s − 9.84·7^-s − 7.91·9^-s − 0.577i·11^-s − 2.31i·13^-s − 26.4i·17^-s + 29.0·19^-s + 40.5i·21^-s − 18.1i·23^-s − 4.45i·27^-s − 17.1i·29^-s + (−17.5 − 25.5i)31^-s − 2.37·33^-s − 21.3i·37^-s − 9.53·39^-s + ⋯

L(s) = 1

− 1.37i·3^-s − 1.40·7^-s − 0.879·9^-s − 0.0524i·11^-s − 0.178i·13^-s − 1.55i·17^-s + 1.53·19^-s + 1.92i·21^-s − 0.791i·23^-s − 0.165i·27^-s − 0.591i·29^-s + (−0.567 − 0.823i)31^-s − 0.0719·33^-s − 0.576i·37^-s − 0.244·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3100$ = $2^{2} \cdot 5^{2} \cdot 31$
Sign:	$-0.567 - 0.823i$
Analytic conductor:	$84.4688$
Root analytic conductor:	$9.19069$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3100} (1301, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3100,\ (\ :1),\ -0.567 - 0.823i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9347400666$
$L(\frac12)$	$\approx$	$0.9347400666$
$L(2)$		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 + (17.5 + 25.5i)T $
good	3	$ 1 + 4.11iT - 9T^{2} $
	7	$ 1 + 9.84T + 49T^{2} $
	11	$ 1 + 0.577iT - 121T^{2} $
	13	$ 1 + 2.31iT - 169T^{2} $
	17	$ 1 + 26.4iT - 289T^{2} $
	19	$ 1 - 29.0T + 361T^{2} $
	23	$ 1 + 18.1iT - 529T^{2} $
	29	$ 1 + 17.1iT - 841T^{2} $
	37	$ 1 + 21.3iT - 1.36e3T^{2} $
	41	$ 1 + 27.2T + 1.68e3T^{2} $
	43	$ 1 + 2.63iT - 1.84e3T^{2} $
	47	$ 1 - 37.5T + 2.20e3T^{2} $
	53	$ 1 + 72.9iT - 2.80e3T^{2} $
	59	$ 1 - 46.3T + 3.48e3T^{2} $
	61	$ 1 + 7.97iT - 3.72e3T^{2} $
	67	$ 1 - 67.9T + 4.48e3T^{2} $
	71	$ 1 + 78.5T + 5.04e3T^{2} $
	73	$ 1 - 128. iT - 5.32e3T^{2} $
	79	$ 1 - 31.8iT - 6.24e3T^{2} $
	83	$ 1 + 50.4iT - 6.88e3T^{2} $
	89	$ 1 - 64.1iT - 7.92e3T^{2} $
	97	$ 1 + 132.T + 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

−7.81479355859300003816354973778, −7.10106071072431046879663436225, −6.80332257848308705887257725353, −5.89995728350455518799943207129, −5.22879454402187389232813556834, −3.89094971653422197318560580072, −2.93477745800009252003146937906, −2.31659150318430402067898588631, −0.920550332573689739026397599596, −0.25535285179927141497753362597, 1.43258368091229497149477252550, 3.02783231637201859152701409771, 3.49632191235717819632590574509, 4.18006493794617848078392213950, 5.21809944264745454122385266507, 5.80466719824007335147109222504, 6.69658833439800890841775688679, 7.46757973562208638017415781793, 8.543274922828699898886500796503, 9.275105389724528969787543346895

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9347400666\)
\(L(\frac12)\)	\(\approx\)	\(0.9347400666\)
\(L(2)\)		not available
\(L(1)\)		not available

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