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Properties

Label 1-2e6-64.29-r0-0-0
Degree $1$
Conductor $64$
Sign $0.995 - 0.0980i$
Analytic cond. $0.297214$
Root an. cond. $0.297214$
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  + (0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)13-s − i·15-s + i·17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (−0.923 − 0.382i)29-s − 31-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)13-s − i·15-s + i·17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (−0.923 − 0.382i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.995 - 0.0980i$
Analytic conductor: \(0.297214\)
Root analytic conductor: \(0.297214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (0:\ ),\ 0.995 - 0.0980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139296592 - 0.05597005255i\)
\(L(\frac12)\) \(\approx\) \(1.139296592 - 0.05597005255i\)
\(L(1)\) \(\approx\) \(1.232080730 + 0.02781530869i\)
\(L(1)\) \(\approx\) \(1.232080730 + 0.02781530869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (-0.382 - 0.923i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 - T \)
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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

−31.72767436844139208738423911267, −31.380345561853704940477793858191, −30.19701022949553849381931632575, −29.35107762511511312731455676498, −27.54094863023883067142901864575, −26.75727318256883526562718742384, −25.605745502505357816358877647958, −24.65338290380527727790769103504, −23.52043346483644352524245125627, −22.14267124240086251764129525029, −20.93647603389952435776181753355, −19.840703973405342560743660432843, −18.43488168960297054128202120169, −18.17069000106866864028836396650, −15.837667224396152007788729301210, −14.90328890730817422428685867451, −13.99123185575273673402839645659, −12.549208560747391273181108350394, −11.203363691771786506561929443241, −9.732696330216818831541813382087, −8.11771473278443569354260868236, −7.426092800465005318973478490809, −5.53845643692668285308686015481, −3.42968428235922852394869272527, −2.27752717834523380918554012100, 1.94793471944494086816269791353, 4.0048916562852408839103641437, 4.9260467942592083105867321299, 7.4253340202776252554441693850, 8.352469010463439010624051097960, 9.60325088205408122465296497427, 10.991264612932908055313704842530, 12.66116385235673894014680508664, 13.772752614546248389703923276805, 14.98264605708834905268156112077, 16.11369456761171188567158556742, 17.26104379380408616490157998275, 18.9134449898731447334631274588, 20.16220939388545075440437868505, 20.70380013162467957166132166112, 21.867930753557755275890635505234, 23.84618068584189719160611871492, 24.20264355956364454234596804088, 25.834189527740095073044408411548, 26.6386513574994404172130068103, 27.745922434936229049540609237191, 28.775081442741461935223219464044, 30.46905657956390289506583176310, 31.15282655854938056820826254279, 32.23277118107529599834800281073

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