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Properties

Label 1-29-29.23-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: $\chi_{29} (23, \cdot )$
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

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

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