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Properties

Label 1-3e4-81.49-r0-0-0
Degree $1$
Conductor $81$
Sign $-0.910 - 0.413i$
Analytic cond. $0.376162$
Root an. cond. $0.376162$
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  + (−0.835 − 0.549i)2-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)5-s + (−0.993 + 0.116i)7-s + (0.173 − 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.286 − 0.957i)11-s + (−0.0581 − 0.998i)13-s + (0.893 + 0.448i)14-s + (−0.686 + 0.727i)16-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.396 − 0.918i)20-s + (−0.286 + 0.957i)22-s + (−0.993 − 0.116i)23-s + ⋯
L(s)  = 1  + (−0.835 − 0.549i)2-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)5-s + (−0.993 + 0.116i)7-s + (0.173 − 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.286 − 0.957i)11-s + (−0.0581 − 0.998i)13-s + (0.893 + 0.448i)14-s + (−0.686 + 0.727i)16-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.396 − 0.918i)20-s + (−0.286 + 0.957i)22-s + (−0.993 − 0.116i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.910 - 0.413i$
Analytic conductor: \(0.376162\)
Root analytic conductor: \(0.376162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ -0.910 - 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06798210101 - 0.3138403109i\)
\(L(\frac12)\) \(\approx\) \(0.06798210101 - 0.3138403109i\)
\(L(1)\) \(\approx\) \(0.4136351502 - 0.2493305003i\)
\(L(1)\) \(\approx\) \(0.4136351502 - 0.2493305003i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.835 - 0.549i)T \)
5 \( 1 + (-0.686 - 0.727i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (-0.0581 - 0.998i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.993 - 0.116i)T \)
29 \( 1 + (0.893 - 0.448i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.835 + 0.549i)T \)
43 \( 1 + (0.973 + 0.230i)T \)
47 \( 1 + (0.597 + 0.802i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (0.396 - 0.918i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.686 + 0.727i)T \)
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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

−31.54831063339679830534870738681, −30.33529620166119830088406879770, −29.0191305489126129029211332766, −28.22953244619765459120191183390, −26.97693483761809744499916647591, −26.12059654519049118309153476625, −25.49434968728473579116015304241, −23.863132124367401562982236092995, −23.19978999548626934476126258566, −21.95295220199889525068398999637, −20.07093457004479889811272235056, −19.373356241302581837113041610205, −18.39893062708316669677148816689, −17.22783617113302728305619513109, −15.923205463813200622971160639317, −15.26400786737337880551995065850, −13.96172734148525854883665449484, −12.21882298983379598689022427278, −10.81235666170689836746666175473, −9.84374336551936755084332419450, −8.51105997545296256905319200027, −7.05836716062445666040158787395, −6.44992524422141171980649764297, −4.33512038480674171973659034811, −2.36489818161407116103996483882, 0.44070899055071802399870327433, 2.7320469218591978380714065530, 4.08978519747750148970540196016, 6.19171157033316016139585040025, 7.86956036234664347033386932963, 8.73701327897348289009940421836, 10.01183982565490881773310172282, 11.25228778597876118922337283352, 12.4725059439973475743504634517, 13.277795150670314572416988789084, 15.605846270918726122143287215450, 16.22190562040864001545873408709, 17.394892676922022271251611791288, 18.77744259488796196103836789810, 19.61930294573911923010275616824, 20.44114130776135883411647200883, 21.70925587192329047478047485877, 22.87099344362361859814835894437, 24.3433750957121932440229569275, 25.35403345023529149776466980397, 26.5500857337169225447282034735, 27.4132133949659565882071815065, 28.43958367541536774043751615938, 29.2502630719260462658675076638, 30.29854492391306981043137606634

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