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Properties

Label 1-29-29.4-r0-0-0
Degree $1$
Conductor $29$
Sign $0.549 + 0.835i$
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  + (0.900 + 0.433i)2-s + (−0.623 + 0.781i)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 + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (0.222 − 0.974i)11-s − 12-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)14-s + (0.900 − 0.433i)15-s + (−0.222 + 0.974i)16-s − 17-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (−0.623 + 0.781i)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 + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (0.222 − 0.974i)11-s − 12-s + (−0.222 + 0.974i)13-s + (0.900 − 0.433i)14-s + (0.900 − 0.433i)15-s + (−0.222 + 0.974i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ 0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8064168362 + 0.4349427181i\)
\(L(\frac12)\) \(\approx\) \(0.8064168362 + 0.4349427181i\)
\(L(1)\) \(\approx\) \(1.077049267 + 0.4215966082i\)
\(L(1)\) \(\approx\) \(1.077049267 + 0.4215966082i\)

Euler product

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

−37.41808679562710906595742203297, −35.65695106660538093102222453802, −34.43174004874618405429626800980, −33.60039512014922029862661760386, −31.71838589647010520321539231934, −30.71538594094610624515204753280, −30.01340767619313303550922960124, −28.44094811384944890252806902727, −27.593786791357491147688863947975, −25.13336245192741277037103808850, −24.12074602330475528342970162447, −22.92136513691032381946661963376, −22.17083273994220280787104593580, −20.29886597665274510999961915206, −19.06958068634007048620198252050, −17.82051430385000077632984769072, −15.661658119750848426988457037172, −14.54307151620837474748488643341, −12.6675824600500515469362466659, −11.86947419608088271795606158009, −10.65827512491927438892554283239, −7.78257271097145570944077656840, −6.22764632750062932444329944213, −4.57779880641955657892633833974, −2.26652633876986064341917098437, 3.89565228477907479947220531468, 4.80508075297458321234305222877, 6.69858851890326554103351427511, 8.52045435512501431876556296179, 11.0422774578459253414106962572, 11.84607752100139660521656917622, 13.74795701017881827758499864155, 15.24025122485106356500518472065, 16.338788478497863814748383285776, 17.27492724744525599172840203380, 19.85312646640535031916530013058, 21.120063817420526569581986324490, 22.20467570591636532778828843419, 23.69724570451140494525850264228, 24.089541859157331374978524098079, 26.368931199950683562494567082405, 27.20172586439005238035809340718, 28.79428994776223965055859810945, 30.22788840255068523099093883084, 31.61257328900231348120035773463, 32.5352189731700745651824410614, 33.718759123761658319183960079573, 34.6002555042892052782182572361, 35.88721001744814935026099853259, 37.94750136114824778965924614058

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