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Properties

Label 1-29-29.19-r1-0-0
Degree $1$
Conductor $29$
Sign $0.995 + 0.0915i$
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  + (−0.433 + 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.974 + 0.222i)11-s + i·12-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.433 − 0.900i)15-s + (−0.222 + 0.974i)16-s − i·17-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (0.900 − 0.433i)6-s + (0.623 − 0.781i)7-s + (0.974 − 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.974 + 0.222i)11-s + i·12-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.433 − 0.900i)15-s + (−0.222 + 0.974i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(29\)
Sign: $0.995 + 0.0915i$
Analytic conductor: \(3.11648\)
Root analytic conductor: \(3.11648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 29,\ (1:\ ),\ 0.995 + 0.0915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036806000 + 0.04756654195i\)
\(L(\frac12)\) \(\approx\) \(1.036806000 + 0.04756654195i\)
\(L(1)\) \(\approx\) \(0.8568338239 + 0.1013611466i\)
\(L(1)\) \(\approx\) \(0.8568338239 + 0.1013611466i\)

Euler product

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

−37.37645670041370420854384355795, −35.83210771044771497461821649231, −34.62698441998043606967371491405, −33.276553473324365551164334191734, −32.007523844569122851497968043323, −30.45736021203227074546160197293, −29.0777791714324372657860602526, −28.30981461887925118996928413111, −27.356328726803946972941983802980, −25.91899522446749659210779286520, −24.2306076385507521874195727213, −22.22169229728953273153179738117, −21.57150718591431718674530546422, −20.52640504900480856883949471653, −18.59836050650690781555393859297, −17.447899638051577674459894302521, −16.47835026237073442330052541516, −14.27105756673622993132773140345, −12.375304923821431352259963061217, −11.34906711451598779326578306977, −9.81658537106823440144192915625, −8.77086558335585713097009621461, −5.88751288896548736915049419477, −4.2129384210276921064335887856, −1.624482192777851356915596144287, 1.206309939527681689256645327432, 5.100224694738397674018041688564, 6.49438789284978729864913647264, 7.646467688411418059686950519492, 9.76231287548769352824883963993, 11.15222924562921701289161606038, 13.38303300051010020536719005665, 14.398275635453950716661499121741, 16.361443465830924156368184224211, 17.70197294184075725435759899942, 18.0067324705171953152122664261, 19.88819554539911183253875080922, 22.14876747301269972330489422089, 23.132056465446898692182859060137, 24.49370962785625693134520127697, 25.33963623871959342561630656119, 26.92760552938856117912223062680, 28.05713835018505960569956812958, 29.49190165863024297267883300812, 30.480061552927352693110962892897, 32.701769640861338063607246612684, 33.50346117751801689106783340171, 34.45299824367336704189971217567, 35.68088718644992583502312664157, 36.6768024750537109595614462980

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