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Properties

Label 1-29-29.24-r0-0-0
Degree $1$
Conductor $29$
Sign $-0.833 + 0.552i$
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  + (−0.222 + 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)11-s + 12-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)11-s + 12-s + (0.623 + 0.781i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(0.134675\)
Root analytic conductor: \(0.134675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (0:\ ),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1252539855 + 0.4153005135i\)
\(L(\frac12)\) \(\approx\) \(0.1252539855 + 0.4153005135i\)
\(L(1)\) \(\approx\) \(0.4060286231 + 0.4387473455i\)
\(L(1)\) \(\approx\) \(0.4060286231 + 0.4387473455i\)

Euler product

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

−36.57927284035358527020775944502, −35.6462944652756531689059482402, −34.91493620124064258561448220371, −32.78775369590765265901536744270, −31.796202841561475843714425398355, −29.99144329817097919310611084697, −29.386163400171109665340805557718, −28.0898412100639471747909975924, −27.38146516208638100938692513765, −25.40734523332958955083670525640, −23.661221899116944224779909940291, −22.71765027143154057888121936589, −21.31904373946677078635544504099, −19.829941058161542155152524800035, −18.81356811761612836733901966176, −17.184078559834183717762828146357, −16.35821936653212835501551419773, −13.41939603830550997037822142997, −12.564350442222234121164453558404, −11.30354467973762113377730018629, −9.81319872187136677468355662618, −8.06265765543259523867381909386, −5.76138533550202578842369012, −3.84734117535889590769822772057, −1.00015516298170237827229722153, 4.08081536101879485562656864049, 6.076938513657141513085772390860, 6.95260585981191078176305950675, 9.27687897191694663018669806762, 10.55565615586021738439767506672, 12.39192319389359990463903614070, 14.47081709930705995664168065457, 15.63383290726376302209832182125, 16.71966022143983979342183067697, 18.11746922850488862886385442705, 19.15677544379112439448877309146, 21.74189713712751517079452276206, 22.76107567693998433608008161269, 23.50134608875226177624719111419, 25.407223089819509606002718526283, 26.36128886257882442925785325811, 27.65166844714059773299516858789, 28.63859256571346564009026616026, 30.37317107927239225701839162902, 32.046577545962595831616672116221, 33.155331828851572117256530086165, 34.19187594451787179787235486982, 35.002027431403321018488499625414, 36.08608405874741486374479895124, 38.099194941985007198342838717

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