Properties

Degree 1
Conductor 29
Sign $-0.647 - 0.761i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.781 − 0.623i)2-s + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (−0.974 − 0.222i)10-s + (−0.433 + 0.900i)11-s + i·12-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.781 + 0.623i)15-s + (−0.900 − 0.433i)16-s i·17-s + ⋯
L(s,χ)  = 1  + (0.781 − 0.623i)2-s + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)6-s + (−0.222 − 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (−0.974 − 0.222i)10-s + (−0.433 + 0.900i)11-s + i·12-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (0.781 + 0.623i)15-s + (−0.900 − 0.433i)16-s i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.647 - 0.761i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.647 - 0.761i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.647 - 0.761i$
motivic weight  =  \(0\)
character  :  $\chi_{29} (11, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 29,\ (1:\ ),\ -0.647 - 0.761i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5266494350 - 1.139198729i$
$L(\frac12,\chi)$  $\approx$  $0.5266494350 - 1.139198729i$
$L(\chi,1)$  $\approx$  0.8558977307 - 0.6589800951i
$L(1,\chi)$  $\approx$  0.8558977307 - 0.6589800951i

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

−37.75543004271375285050490415837, −35.30071555037263780982302470264, −34.900777062249056678366853677187, −33.93480192219335289084224490106, −32.62276320734219820817784323730, −31.16595342434195412651204459483, −30.22191491820980780987429600948, −28.84633617909296494543043882995, −27.30089912457735035819772456564, −25.88304240476270323785155702456, −24.401221291309547870210537051515, −23.338596580508229293963863142366, −22.34911704887705895653853065142, −21.39779386405463253563192193826, −18.96783727437158316341507204701, −17.79102298864331363930172849290, −16.06943386015238726808577491729, −15.32929421984563181146381637221, −13.43364710149578146076085127280, −12.02926139921453587514462118083, −10.9633172903835575438091565976, −8.14011235199247140640569741096, −6.53395156253449730742010663178, −5.43731734385044667128186961082, −3.31077422201071744138994531220, 0.857889842621212458023636053494, 3.99371708555228898890407169288, 5.096640965904074460604176472198, 6.997427609075130594386528527654, 9.75782432156971915785778251596, 11.14091081738288091238599689853, 12.27845707813199937712039662615, 13.50487667362348920446696271045, 15.55333091730257643749158749732, 16.58297853303267306644327736502, 18.4256850190098632957672223572, 20.19517012155017227546697700064, 20.97008970442137623631343409255, 22.86071408752858060385315494963, 23.243285180097225294197428872438, 24.52045595003607325659052272532, 26.87949788749073532592728646119, 28.20852030625269360639138135458, 28.89762786909636024246637396006, 30.27073544681008763253625810393, 31.54291645923619831869373464904, 32.94109717368958071867304469414, 33.56174510700917983452622234682, 35.40518458118424502533895766776, 36.46670722350424149668306399740

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