Properties

Label 1-29-29.10-r1-0-0
Degree $1$
Conductor $29$
Sign $0.715 - 0.698i$
Analytic cond. $3.11648$
Root an. cond. $3.11648$
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  + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s i·12-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s + i·17-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.781 − 0.623i)10-s + (−0.974 − 0.222i)11-s i·12-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (0.433 + 0.900i)15-s + (−0.222 + 0.974i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.715 - 0.698i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ 0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.019526541 - 0.8219827192i\)
\(L(\frac12)\) \(\approx\) \(2.019526541 - 0.8219827192i\)
\(L(1)\) \(\approx\) \(1.607775250 - 0.5051594150i\)
\(L(1)\) \(\approx\) \(1.607775250 - 0.5051594150i\)

Euler product

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

−36.62206442634949235245163788668, −36.14000881168350163576523553163, −34.50000054051843011200847399237, −33.478261346335069698301720203178, −31.98318786162667095218734398665, −31.295303298256065065819407533168, −30.01747036729571956083038243042, −28.408289135711714958968182789596, −26.47143978210190184320581019194, −25.469005199037844790533722915197, −24.57250497752782310610425371287, −23.616881345228886736821371650429, −21.61352561586080626114205134354, −20.76948549474588535844032025366, −18.52788973054692222833966804559, −17.68173701727030892970621120911, −15.89040691336259250312144215255, −14.44829687946804026573966876336, −13.48026980732169470106766841378, −12.25367715325095036270516538500, −9.264223853844620541376941805765, −8.21062628537667045847387786286, −6.53123332195502238356460766533, −4.90641606986875696017892434199, −2.37695957689145852606659833082, 2.105384994079940654353416695618, 3.75069900661620720258363749484, 5.490326561383460650308063631744, 8.25947504168891974121934927231, 10.17015652356203222584299064695, 10.657797171874414295373478706463, 13.12162943458100624006570019854, 14.05246429032412334651953518349, 15.2142241317931708151866077725, 17.48767435345017586633011991631, 18.96065385844321962134594341719, 20.49637552452733423484022592001, 21.14006577684221171685518531273, 22.335333892828165735875678951393, 23.91557932122032153280770622949, 25.68584763147432107595705059387, 26.8497754041585174163435036840, 28.076983326652216398797951535798, 29.69483416569308317921767748891, 30.46878692734011666661714303969, 31.832162750035468301585919084421, 32.90204232055572251251634565273, 33.810627037120124508961472958701, 36.3887787775696410616030558024, 37.12576164201851730220604498195

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