Properties

Label 1-29-29.23-r0-0-0
Degree 11
Conductor 2929
Sign 0.8330.552i-0.833 - 0.552i
Analytic cond. 0.1346750.134675
Root an. cond. 0.1346750.134675
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(29s/2ΓR(s)L(s)=((0.8330.552i)Λ(1s)\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}
Λ(s)=(29s/2ΓR(s)L(s)=((0.8330.552i)Λ(1s)\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: 11
Conductor: 2929
Sign: 0.8330.552i-0.833 - 0.552i
Analytic conductor: 0.1346750.134675
Root analytic conductor: 0.1346750.134675
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ29(23,)\chi_{29} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 29, (0: ), 0.8330.552i)(1,\ 29,\ (0:\ ),\ -0.833 - 0.552i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.12525398550.4153005135i0.1252539855 - 0.4153005135i
L(12)L(\frac12) \approx 0.12525398550.4153005135i0.1252539855 - 0.4153005135i
L(1)L(1) \approx 0.40602862310.4387473455i0.4060286231 - 0.4387473455i
L(1)L(1) \approx 0.40602862310.4387473455i0.4060286231 - 0.4387473455i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad29 1 1
good2 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
3 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
5 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
7 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
11 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
13 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
17 1+T 1 + T
19 1+(0.900+0.433i)T 1 + (-0.900 + 0.433i)T
23 1+(0.222+0.974i)T 1 + (-0.222 + 0.974i)T
31 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
37 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
41 1+T 1 + T
43 1+(0.222+0.974i)T 1 + (-0.222 + 0.974i)T
47 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
53 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
59 1+T 1 + T
61 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
67 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
71 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
73 1+(0.222+0.974i)T 1 + (-0.222 + 0.974i)T
79 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
83 1+(0.900+0.433i)T 1 + (-0.900 + 0.433i)T
89 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
97 1+(0.900+0.433i)T 1 + (-0.900 + 0.433i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line