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Properties

Label 1-29-29.20-r0-0-0
Degree $1$
Conductor $29$
Sign $0.855 - 0.517i$
Analytic cond. $0.134675$
Root an. cond. $0.134675$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)5-s + (0.623 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)11-s + 12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + 17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)5-s + (0.623 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)11-s + 12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8670025821 - 0.2419789471i\)
\(L(\frac12)\) \(\approx\) \(0.8670025821 - 0.2419789471i\)
\(L(1)\) \(\approx\) \(1.107935304 - 0.2465575357i\)
\(L(1)\) \(\approx\) \(1.107935304 - 0.2465575357i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−36.94961694925969965086927833969, −36.07784148589745953785555874544, −34.54052223603958905751855953794, −33.97439334764516360642997561299, −32.565642483692090955492166232507, −31.15216951447500548655986026265, −29.87422545017352444003071125672, −29.34434995150138709418098390670, −26.83677781008153409066475554693, −25.75999515109237803186809704530, −24.654817954722246583627492623241, −23.33703394391592055027966476857, −22.58576655421226264164138668789, −20.959154739538277713774750518522, −18.93799746367965548045941164598, −17.643664362639249722445539823171, −16.617693794563357237382129598645, −14.56242637685005727140846513990, −13.63032461782041315813390405205, −12.449968794676153735961084861045, −10.478710533679796669406020076019, −7.89168470411729630356460724305, −6.84017801857655669265959072418, −5.49183744252799304859137727700, −2.95339600789876898960738874757, 2.61651033357808942884242111086, 4.76049809736198473232714243615, 5.66282989807631649012267817691, 9.14032345328068345315784728183, 10.02645301479179505306856722472, 11.74763329073383371968176783727, 12.96277523568800026875007687731, 14.69076706307787309494151081602, 15.91086718365607048026202776731, 17.62570589298972592738392630703, 19.42733449376387309090769850350, 20.99954262032163585143748860372, 21.463178604150063307236876344128, 22.75725135996340718917740299621, 24.25221451545722138060684098177, 25.83854410020353207462493594906, 27.71427759350211912667125558644, 28.46364549863046286890333186446, 29.38177252567326692571952768931, 31.43322853231579063423886782330, 32.00319791325642011574843912590, 33.17066664678966439088990420063, 34.32700190206809129728835216859, 36.49248713724828751864705575186, 37.39027899654660773251304173249

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