Properties

Label 2-21-21.5-c11-0-6
Degree $2$
Conductor $21$
Sign $0.603 - 0.797i$
Analytic cond. $16.1352$
Root an. cond. $4.01686$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−364.5 + 210. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−3.85e4 − 2.21e4i)7-s + (8.85e4 − 1.53e5i)9-s + (7.46e5 + 4.30e5i)12-s + 1.22e6i·13-s + (−2.09e6 + 3.63e6i)16-s + (1.26e7 + 7.28e6i)19-s + (1.87e7 − 5.63e4i)21-s + (2.44e7 + 4.22e7i)25-s + 7.45e7i·27-s + (2.74e5 + 9.10e7i)28-s + (1.01e8 − 5.85e7i)31-s − 3.62e8·36-s + (3.31e8 − 5.74e8i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.867 − 0.497i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + 0.911i·13-s + (−0.499 + 0.866i)16-s + (1.16 + 0.675i)19-s + (0.999 − 0.00301i)21-s + (0.499 + 0.866i)25-s + 0.999i·27-s + (0.00301 + 0.999i)28-s + (0.636 − 0.367i)31-s − 36-s + (0.786 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(16.1352\)
Root analytic conductor: \(4.01686\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :11/2),\ 0.603 - 0.797i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.708006 + 0.352308i\)
\(L(\frac12)\) \(\approx\) \(0.708006 + 0.352308i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (364.5 - 210. i)T \)
7 \( 1 + (3.85e4 + 2.21e4i)T \)
good2 \( 1 + (1.02e3 + 1.77e3i)T^{2} \)
5 \( 1 + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 - 1.22e6iT - 1.79e12T^{2} \)
17 \( 1 + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (-1.26e7 - 7.28e6i)T + (5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 1.22e16T^{2} \)
31 \( 1 + (-1.01e8 + 5.85e7i)T + (1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-3.31e8 + 5.74e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 5.50e17T^{2} \)
43 \( 1 + 1.76e9T + 9.29e17T^{2} \)
47 \( 1 + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (-1.07e10 - 6.21e9i)T + (2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-1.07e10 - 1.85e10i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 - 2.31e20T^{2} \)
73 \( 1 + (-2.13e9 + 1.23e9i)T + (1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (1.07e10 - 1.85e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + 1.28e21T^{2} \)
89 \( 1 + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 1.26e11iT - 7.15e21T^{2} \)
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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

−15.87120685711821636619520165145, −14.48745541171504812652862850540, −13.14209106355966646318866666612, −11.52488076636173629532336542752, −10.17615987994714299937514811430, −9.354901834655399639726450017452, −6.76196309098378319371975946692, −5.44999995668248908107810621960, −3.95667666988824479989651290073, −0.984079518132159641243900442276, 0.47020608590198100993697020433, 2.99152536230134208411209198025, 5.05403879296783702763291689479, 6.69072964154620511027888855633, 8.183724634892179577670966637019, 9.886125155080246848289007398095, 11.68185929555701624873536147365, 12.66519505913501434828279708948, 13.54625619543939111656750465412, 15.70239018114830833869061258745

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