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Properties

Label 1-147-147.53-r1-0-0
Degree $1$
Conductor $147$
Sign $-0.972 - 0.232i$
Analytic cond. $15.7973$
Root an. cond. $15.7973$
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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.826 + 0.563i)5-s + (0.900 + 0.433i)8-s + (0.826 + 0.563i)10-s + (0.988 + 0.149i)11-s + (0.623 − 0.781i)13-s + (0.0747 − 0.997i)16-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (0.222 − 0.974i)20-s + (−0.222 − 0.974i)22-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (−0.826 + 0.563i)5-s + (0.900 + 0.433i)8-s + (0.826 + 0.563i)10-s + (0.988 + 0.149i)11-s + (0.623 − 0.781i)13-s + (0.0747 − 0.997i)16-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (0.222 − 0.974i)20-s + (−0.222 − 0.974i)22-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.972 - 0.232i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ -0.972 - 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05878677629 - 0.4978262999i\)
\(L(\frac12)\) \(\approx\) \(0.05878677629 - 0.4978262999i\)
\(L(1)\) \(\approx\) \(0.5908235393 - 0.2632459326i\)
\(L(1)\) \(\approx\) \(0.5908235393 - 0.2632459326i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.826 - 0.563i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−28.00891750156437288812994014361, −27.423521530865820878252784236150, −26.40754968868966936500296389326, −25.46394501021138190413609261179, −24.29011666539683688250958731636, −23.842242919355458080576274919413, −22.746396832595037972804980471680, −21.69023831398639001709963103297, −20.02641948187355484417275655619, −19.44239913848473613957479756493, −18.25861440059490049724572648401, −17.17059525805302181600072811332, −16.22510459622805847768437037710, −15.53415619270096233951064188721, −14.34356916854443694326828487993, −13.33333083079484718115100892608, −11.941612148700246621946038468953, −10.80318300659946167860429346763, −9.082607363756476108340411908690, −8.69563832590366999480267572992, −7.26750415486846815340750805130, −6.353965240346305073463714328183, −4.812595637567040127939080234209, −3.891243885952827454311480139611, −1.34733067230112283382461488552, 0.23760451056943978706813090379, 1.98879850079978464121588569094, 3.51896698087857134217100533972, 4.27678530731301281540904640712, 6.28988277852378942137353181020, 7.77090173327496511965340611419, 8.644816968067583924101201848227, 10.01691821626512782599823522644, 11.0010758800352197262722828046, 11.84221214993597411060928545641, 12.86406347958250127200812368122, 14.14481125843715480774723879835, 15.27548152490913922908332377298, 16.56881208314179384913517701057, 17.74694751969325689078425625330, 18.60944961687666605353308725761, 19.640701410906589466307975266290, 20.22038289762334218166885136142, 21.5041468733076323140662005215, 22.58021330243362892444148373919, 23.05930595623157771190697130883, 24.58874943963192474966496791795, 25.87920633377894809487066043024, 26.71941344783814783618306233514, 27.66468144682311158939723128621

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