L-function 1-1-1.1-r0-0-0: Riemann zeta function

• Label: 1-1-1.1-r0-0-0
• Degree: $1$
• Conductor: $1$
• Sign: $1$
• Analytic cond.: $0.00464398$
• Root an. cond.: $0.00464398$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

The Riemann zeta function is the prototypical L-function. It is the only L-function of degree 1 and conductor 1, and (conjecturally) it is the only primitive L-function with a pole. Its unique pole is located at $s=1$. Learn more about its history.

Dirichlet series

L(s)  = 1

+ 2^-s + 3^-s + 4^-s + 5^-s + 6^-s + 7^-s + 8^-s + 9^-s + 10^-s + 11^-s + 12^-s + 13^-s + 14^-s + 15^-s + 16^-s + 17^-s + 18^-s + 19^-s + 20^-s + 21^-s + 22^-s + 23^-s + 24^-s + 25^-s + 26^-s + 27^-s + 28^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1$
Sign:	$1$
Analytic conductor:	$0.00464398$
Root analytic conductor:	$0.00464398$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(1,\ 1,\ (0:\ ),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-1.460354508$
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
good	2	$1 - T$
	3	$1 - T$
	5	$1 - T$
	7	$1 - T$
	11	$1 - T$
	13	$1 - T$
	17	$1 - T$
	19	$1 - T$
	23	$1 - T$

Imaginary part of the first few zeros on the critical line

−88.809111207634465423682348079510, −87.425274613125229406531667850919, −84.735492980517050105735311206828, −82.910380854086030183164837494771, −79.337375020249367922763592877116, −77.14484006887480537268266485631, −75.70469069908393316832691676203, −72.06715767448190758252210796983, −69.54640171117397925292685752656, −67.07981052949417371447882889652, −65.11254404808160666087505425318, −60.83177852460980984425990182452, −59.34704400260235307965364867499, −56.44624769706339480436775947671, −52.97032147771446064414729660888, −49.77383247767230218191678467856, −48.00515088116715972794247274943, −43.32707328091499951949612216541, −40.91871901214749518739812691463, −37.58617815882567125721776348071, −32.93506158773918969066236896407, −30.42487612585951321031189753058, −25.01085758014568876321379099256, −21.02203963877155499262847959390, −14.13472514173469379045725198356, 14.13472514173469379045725198356, 21.02203963877155499262847959390, 25.01085758014568876321379099256, 30.42487612585951321031189753058, 32.93506158773918969066236896407, 37.58617815882567125721776348071, 40.91871901214749518739812691463, 43.32707328091499951949612216541, 48.00515088116715972794247274943, 49.77383247767230218191678467856, 52.97032147771446064414729660888, 56.44624769706339480436775947671, 59.34704400260235307965364867499, 60.83177852460980984425990182452, 65.11254404808160666087505425318, 67.07981052949417371447882889652, 69.54640171117397925292685752656, 72.06715767448190758252210796983, 75.70469069908393316832691676203, 77.14484006887480537268266485631, 79.337375020249367922763592877116, 82.910380854086030183164837494771, 84.735492980517050105735311206828, 87.425274613125229406531667850919, 88.809111207634465423682348079510

Graph of the $Z$-function along the critical line

The first zero of the Riemann zeta function, at height approximately 14.134, is higher than that of any other algebraic L-function.