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Properties

Label 1-147-147.110-r0-0-0
Degree $1$
Conductor $147$
Sign $0.232 + 0.972i$
Analytic cond. $0.682665$
Root an. cond. $0.682665$
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

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

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(0.682665\)
Root analytic conductor: \(0.682665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (0:\ ),\ 0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7949167602 + 0.6270062466i\)
\(L(\frac12)\) \(\approx\) \(0.7949167602 + 0.6270062466i\)
\(L(1)\) \(\approx\) \(0.8678306477 + 0.4480746603i\)
\(L(1)\) \(\approx\) \(0.8678306477 + 0.4480746603i\)

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

−28.2219125050443935765574022136, −27.174503364626530774621612462077, −26.155721518861717019050303403971, −25.17493283973188781365481286810, −24.13454712514278494710548179262, −22.676466154340013281447496429702, −21.79170101884941345870909638301, −21.036245989814255068646751062804, −20.00911811399630909396647431677, −19.177736333640537952042496905441, −17.92062446000415982722175696315, −17.164208103771430725072323063851, −16.28350238235238069376697450343, −14.32119317495646900143160126338, −13.64239645522713924075440140115, −12.31443279149701070092520483237, −11.68931406838868187177269548004, −10.10007187698563378535719418096, −9.4600948778805989622603693591, −8.455158048188150825601010489523, −6.90569065099987178962886131151, −5.246783344384589579098441992753, −4.08596402816687410128093695931, −2.47889155401887718848802708006, −1.25675458029995377066440785353, 1.54594098500521725537950964826, 3.5168540512115621734321349695, 5.27585578125648669928353309106, 6.14877866947752247106809240367, 7.238917180786045276468885668818, 8.40912458693082298889439663741, 9.77858151970309058864713597782, 10.28158401564541356109126095427, 12.02653783279254642696978842178, 13.513561247124444062851591665680, 14.37595939872846236768748758395, 15.098082140512789501136447180031, 16.539454632634465707480834383220, 17.26741625374446984445294276673, 18.2127678463466166059088772203, 19.09863592265809397058356826541, 20.30524926379638437046659584187, 21.90283971812592992774440050465, 22.44343885372291783461906394320, 23.60058197690639533463330509199, 24.84659875525520280795064494737, 25.295011613602381415965614355740, 26.32859831306331552446131248886, 27.25809345095859586620577149062, 28.08731757042527975379945663505

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