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Properties

Label 2-21-3.2-c8-0-15
Degree $2$
Conductor $21$
Sign $0.122 - 0.992i$
Analytic cond. $8.55495$
Root an. cond. $2.92488$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.0i·2-s + (9.91 − 80.3i)3-s − 532.·4-s + 850. i·5-s + (−2.25e3 − 278. i)6-s − 907.·7-s + 7.77e3i·8-s + (−6.36e3 − 1.59e3i)9-s + 2.38e4·10-s − 2.11e4i·11-s + (−5.28e3 + 4.28e4i)12-s − 2.10e4·13-s + 2.54e4i·14-s + (6.83e4 + 8.43e3i)15-s + 8.19e4·16-s + 9.19e4i·17-s + ⋯
L(s)  = 1  − 1.75i·2-s + (0.122 − 0.992i)3-s − 2.08·4-s + 1.36i·5-s + (−1.74 − 0.214i)6-s − 0.377·7-s + 1.89i·8-s + (−0.970 − 0.242i)9-s + 2.38·10-s − 1.44i·11-s + (−0.254 + 2.06i)12-s − 0.736·13-s + 0.663i·14-s + (1.35 + 0.166i)15-s + 1.25·16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.122 - 0.992i$
Analytic conductor: \(8.55495\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :4),\ 0.122 - 0.992i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.424790 + 0.375616i\)
\(L(\frac12)\) \(\approx\) \(0.424790 + 0.375616i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.91 + 80.3i)T \)
7 \( 1 + 907.T \)
good2 \( 1 + 28.0iT - 256T^{2} \)
5 \( 1 - 850. iT - 3.90e5T^{2} \)
11 \( 1 + 2.11e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.10e4T + 8.15e8T^{2} \)
17 \( 1 - 9.19e4iT - 6.97e9T^{2} \)
19 \( 1 + 7.16e4T + 1.69e10T^{2} \)
23 \( 1 + 4.03e5iT - 7.83e10T^{2} \)
29 \( 1 + 6.81e5iT - 5.00e11T^{2} \)
31 \( 1 - 2.50e5T + 8.52e11T^{2} \)
37 \( 1 + 2.29e6T + 3.51e12T^{2} \)
41 \( 1 - 2.08e6iT - 7.98e12T^{2} \)
43 \( 1 - 9.01e5T + 1.16e13T^{2} \)
47 \( 1 + 6.75e6iT - 2.38e13T^{2} \)
53 \( 1 + 2.49e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.79e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.32e7T + 1.91e14T^{2} \)
67 \( 1 - 7.45e6T + 4.06e14T^{2} \)
71 \( 1 - 4.50e6iT - 6.45e14T^{2} \)
73 \( 1 - 2.57e7T + 8.06e14T^{2} \)
79 \( 1 + 3.52e7T + 1.51e15T^{2} \)
83 \( 1 + 4.65e7iT - 2.25e15T^{2} \)
89 \( 1 + 9.59e6iT - 3.93e15T^{2} \)
97 \( 1 - 3.70e7T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.60645006201963302279134588698, −13.60362466358606140445182373585, −12.45130437114200353873532536684, −11.24417414225622090049024778448, −10.31853794191123573830297090793, −8.466124402811132067291501935541, −6.41553006447900333663141479703, −3.36998077231531144356120876572, −2.28586334828289142845423921858, −0.27146649468249929238416145821, 4.50897920892353549807366120346, 5.32726399076499545910784913799, 7.37337159994788843578025050158, 8.928192571363036917782087482338, 9.677868330065918932112852967041, 12.42559481231165248868031999158, 13.89675743895694192060391258247, 15.20835172772180365754849887842, 15.94086507921211478516201932524, 16.90572797124483989470570557185

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