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Properties

Label 1-29-29.8-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: $\chi_{29} (8, \cdot )$
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

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

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