LMFDB - L-function 1-156-156.47-r1-0-0

Dirichlet series

L(s) = 1

− i·5^-s + i·7^-s − i·11^-s + 17^-s − i·19^-s − 23^-s − 25^-s − 29^-s − i·31^-s + 35^-s − i·37^-s − i·41^-s + 43^-s − i·47^-s − 49^-s + ⋯

L(s) = 1

− i·5^-s + i·7^-s − i·11^-s + 17^-s − i·19^-s − 23^-s − 25^-s − 29^-s − i·31^-s + 35^-s − i·37^-s − i·41^-s + 43^-s − i·47^-s − 49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$1$
Conductor:	$156$ = $2^{2} \cdot 3 \cdot 13$
Sign:	$-0.289 - 0.957i$
Analytic conductor:	$16.7645$
Root analytic conductor:	$16.7645$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{156} (47, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 156,\ (1:\ ),\ -0.289 - 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7905489547 - 1.065349498i$
$L(\frac12)$	$\approx$	$0.7905489547 - 1.065349498i$
$L(1)$	$\approx$	$0.9629445133 - 0.2915561391i$
$L(1)$	$\approx$	$0.9629445133 - 0.2915561391i$

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	13	$ 1 $
good	5	$ 1 $
	7	$ 1 $
	11	$ 1 $
	17	$ 1 $
	19	$ 1 + iT $
	23	$ 1 $
	29	$ 1 $
	31	$ 1 $
	37	$ 1 - iT $
	41	$ 1 $
	43	$ 1 $
	47	$ 1 $
	53	$ 1 $
	59	$ 1 $
	61	$ 1 + T $
	67	$ 1 $
	71	$ 1 - iT $
	73	$ 1 $
	79	$ 1 $
	83	$ 1 $
	89	$ 1 - T $
	97	$ 1 $
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$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−27.79564157912691555825504718399, −26.96030373026663149207982039523, −25.99019119690539952334912893419, −25.348264954989370445275522188159, −23.83560862153370405992837407663, −23.02755941793886884233970574127, −22.33274110371869209560765669939, −21.00657954608994555208235565375, −20.122122683751825305711990038561, −19.026510287051759937863753406924, −18.05818060880996978256683002461, −17.11105978464597943396769031542, −15.97722096760541939781661008813, −14.65461410400428289673673393932, −14.12644367150559723128764425694, −12.767811652812311811527990662603, −11.58598707266241941439331337782, −10.34991617817862654786516662247, −9.82220747076534186215505132982, −7.89207187406111397493493820308, −7.16377641453551323797689012228, −5.96278386935602555327514200489, −4.320022370523815341672545757623, −3.21631633378325478307967088815, −1.59910775196490900489612011329, 0.498159412106850487793084297622, 2.157020363890858921805921681585, 3.72932571653588901004922412823, 5.24847680067723015228663918431, 5.97650688912942371433505584158, 7.78389717459136064339750221154, 8.77047962755021680381268702727, 9.59308248953805830577693708282, 11.207364643840820815153053628241, 12.15965601682234077438328471713, 13.08632629856897314912676918400, 14.230082374996387668715940077539, 15.55532578949516553595397888065, 16.31674645082879068141207859943, 17.34830985058547583892854955872, 18.57301916997727361300071338855, 19.42892780387700475110256073061, 20.61471235760106561016425434706, 21.47390024874872394672942764364, 22.316752708775455113140928503232, 23.81788599512921900410089217029, 24.380660703840901600526181669065, 25.334288116201826993633866102645, 26.36003295749242935448144052867, 27.77562987153526294087777853142

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7905489547 - 1.065349498i\)
\(L(\frac12)\)	\(\approx\)	\(0.7905489547 - 1.065349498i\)
\(L(1)\)	\(\approx\)	\(0.9629445133 - 0.2915561391i\)
\(L(1)\)	\(\approx\)	\(0.9629445133 - 0.2915561391i\)

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