Properties

Label 1-29-29.28-r0-0-0
Degree $1$
Conductor $29$
Sign $1$
Analytic cond. $0.134675$
Root an. cond. $0.134675$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4657396651\)
\(L(\frac12)\) \(\approx\) \(0.4657396651\)
\(L(1)\) \(\approx\) \(0.6117662895\)
\(L(1)\) \(\approx\) \(0.6117662895\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
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 \( 1 + T \)
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

−37.14660426045612608592774008930, −36.13025701533360897988206562729, −34.759097299338730812239604825148, −33.713290308146147432370245351397, −33.12082197101496094180150219340, −30.58932128259149606033586979164, −29.354382967616109726243743295460, −28.51519021063355365846929501319, −27.4535382830190679798435094436, −26.08155139357491946652948772394, −24.698129503089230104650978934238, −23.547316535114245239312127860119, −21.53989598716732091875171728310, −20.75659856447482979112713688701, −18.50778885467398393360472019015, −17.81102604346635775963157582321, −16.78312280030606466990182922981, −15.292358015439331185195448880791, −13.07439942649190669490767093566, −11.19518919966302935918251101493, −10.41960537862811549543736767944, −8.6305478601229218895353492512, −6.75798544539269780881062624194, −5.316636251071941922447217297715, −1.79380926972434260937039337547, 1.79380926972434260937039337547, 5.316636251071941922447217297715, 6.75798544539269780881062624194, 8.6305478601229218895353492512, 10.41960537862811549543736767944, 11.19518919966302935918251101493, 13.07439942649190669490767093566, 15.292358015439331185195448880791, 16.78312280030606466990182922981, 17.81102604346635775963157582321, 18.50778885467398393360472019015, 20.75659856447482979112713688701, 21.53989598716732091875171728310, 23.547316535114245239312127860119, 24.698129503089230104650978934238, 26.08155139357491946652948772394, 27.4535382830190679798435094436, 28.51519021063355365846929501319, 29.354382967616109726243743295460, 30.58932128259149606033586979164, 33.12082197101496094180150219340, 33.713290308146147432370245351397, 34.759097299338730812239604825148, 36.13025701533360897988206562729, 37.14660426045612608592774008930

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