LMFDB - L-function 2-3680-92.91-c1-0-81

Dirichlet series

L(s) = 1

− 1.89i·3^-s + i·5^-s + 4.39·7^-s − 0.595·9^-s + 5.84·11^-s + 4.29·13^-s + 1.89·15^-s − 6.96i·17^-s − 2.11·19^-s − 8.32i·21^-s + (−4.51 − 1.60i)23^-s − 25^-s − 4.55i·27^-s − 3.36·29^-s + 8.00i·31^-s + ⋯

L(s) = 1

− 1.09i·3^-s + 0.447i·5^-s + 1.65·7^-s − 0.198·9^-s + 1.76·11^-s + 1.19·13^-s + 0.489·15^-s − 1.68i·17^-s − 0.484·19^-s − 1.81i·21^-s + (−0.942 − 0.334i)23^-s − 0.200·25^-s − 0.877i·27^-s − 0.624·29^-s + 1.43i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3680$ = $2^{5} \cdot 5 \cdot 23$
Sign:	$0.429 + 0.902i$
Analytic conductor:	$29.3849$
Root analytic conductor:	$5.42078$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3680} (1471, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3680,\ (\ :1/2),\ 0.429 + 0.902i)$

Particular Values

$L(1)$	$\approx$	$2.915776859$
$L(\frac12)$	$\approx$	$2.915776859$
$L(\frac{3}{2})$		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 - iT $
	23	$ 1 + (4.51 + 1.60i)T $
good	3	$ 1 + 1.89iT - 3T^{2} $
	7	$ 1 - 4.39T + 7T^{2} $
	11	$ 1 - 5.84T + 11T^{2} $
	13	$ 1 - 4.29T + 13T^{2} $
	17	$ 1 + 6.96iT - 17T^{2} $
	19	$ 1 + 2.11T + 19T^{2} $
	29	$ 1 + 3.36T + 29T^{2} $
	31	$ 1 - 8.00iT - 31T^{2} $
	37	$ 1 + 0.681iT - 37T^{2} $
	41	$ 1 - 7.33T + 41T^{2} $
	43	$ 1 - 2.93T + 43T^{2} $
	47	$ 1 - 4.08iT - 47T^{2} $
	53	$ 1 + 4.41iT - 53T^{2} $
	59	$ 1 + 8.67iT - 59T^{2} $
	61	$ 1 - 5.30iT - 61T^{2} $
	67	$ 1 - 8.04T + 67T^{2} $
	71	$ 1 - 11.7iT - 71T^{2} $
	73	$ 1 + 15.2T + 73T^{2} $
	79	$ 1 + 15.3T + 79T^{2} $
	83	$ 1 + 6.75T + 83T^{2} $
	89	$ 1 - 11.8iT - 89T^{2} $
	97	$ 1 - 15.2iT - 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

−8.342182169491133716640348806735, −7.55218720652501341665821514534, −6.97231576844384042215701140022, −6.37753884795960848827563958458, −5.55405884844037524589505794461, −4.46808492614429444439017586738, −3.86298008796036537759773782164, −2.52876250645752838017908328510, −1.58511599636523760483231823661, −1.05459661599739029948600607533, 1.33868710894242370690165556942, 1.83300582700311951686691410928, 3.71078750377491409135469530684, 4.17097774136884013404293728296, 4.46665806225845357869838775242, 5.79845251797046929159117792374, 6.02938241136471637738595999823, 7.34894703447214458905150610743, 8.188295687038982372713435316871, 8.723936360990665966749802927869

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.915776859\)
\(L(\frac12)\)	\(\approx\)	\(2.915776859\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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