Properties

Degree 1
Conductor 29
Sign $-0.833 + 0.552i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.833 + 0.552i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (24, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (0:\ ),\ -0.833 + 0.552i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1252539855 + 0.4153005135i$
$L(\frac12,\chi)$  $\approx$  $0.1252539855 + 0.4153005135i$
$L(\chi,1)$  $\approx$  0.4060286231 + 0.4387473455i
$L(1,\chi)$  $\approx$  0.4060286231 + 0.4387473455i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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