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Properties

Label 1-63-63.31-r1-0-0
Degree $1$
Conductor $63$
Sign $0.975 - 0.220i$
Analytic cond. $6.77029$
Root an. cond. $6.77029$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + 11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(6.77029\)
Root analytic conductor: \(6.77029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (1:\ ),\ 0.975 - 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009068532 - 0.1126300311i\)
\(L(\frac12)\) \(\approx\) \(1.009068532 - 0.1126300311i\)
\(L(1)\) \(\approx\) \(0.7721251257 - 0.1745403663i\)
\(L(1)\) \(\approx\) \(0.7721251257 - 0.1745403663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−32.27234367388550506089539956400, −31.28834014941959263804163612217, −29.96758891475449991993986197236, −28.39582314837269400889392274829, −27.39011302290605991398353113837, −26.82796670935613978565861191717, −25.28726045518376063153938644955, −24.57128116719941385609681456635, −23.13020106327546438706788430997, −22.645981026903088723512508396852, −20.5249432869271806590231217374, −19.39429148102674447894349654389, −18.44498028951957130439477699101, −17.090190132324614371653558809257, −16.020463689615229918737142828277, −15.06239546133681930643429028589, −13.88891044009185368615955943932, −12.141960796503782292879499281509, −10.72877701622394508163292131764, −9.21788929630966320384634029624, −8.014996026395983674002173075677, −6.94083968489108383640188101493, −5.36675548432417486792891272818, −3.73331291513385510651575210250, −0.83327325022601813634542349622, 1.208219090186543188809201471520, 3.29993053263841558784128284112, 4.45056607310803967734795079924, 6.92713913626343929795325898470, 8.33530949887168361735404759706, 9.41140556084223642032906292863, 11.075908139655417464021079651530, 11.784747031979044447980676607, 13.066607361733385593164451821340, 14.62926047122161226357683094185, 16.223381223867594002329689963103, 17.262323244442977340349652932302, 18.78398123232665122131872388212, 19.46621528464890729997444958648, 20.54365964799939207609456838974, 21.80226343101008922835716030234, 22.88302897494825072961120702835, 24.11951405073786307700315063247, 25.75783157583616088467880602596, 26.76427788999370105949447346487, 27.76285269568187282247089220934, 28.51927555600703961791802260664, 29.95905032083069313855805417291, 30.76613434959428641946046209356, 31.663374751139834458388274007103

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