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Properties

Label 1-29-29.11-r1-0-0
Degree $1$
Conductor $29$
Sign $-0.647 - 0.761i$
Analytic cond. $3.11648$
Root an. cond. $3.11648$
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  + (0.781 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $-0.647 - 0.761i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ -0.647 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5266494350 - 1.139198729i\)
\(L(\frac12)\) \(\approx\) \(0.5266494350 - 1.139198729i\)
\(L(1)\) \(\approx\) \(0.8558977307 - 0.6589800951i\)
\(L(1)\) \(\approx\) \(0.8558977307 - 0.6589800951i\)

Euler product

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

−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