Properties

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 $0$

Origins

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Normalization:  

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 $s=1$.

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 + ⋯
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

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.460354508\)
\(L(\frac12)\) \(\approx\) \(-1.460354508\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−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.