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Properties

Label 2-21-21.2-c4-0-1
Degree $2$
Conductor $21$
Sign $0.905 - 0.424i$
Analytic cond. $2.17076$
Root an. cond. $1.47335$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.14 − 2.39i)2-s + (2.75 + 8.56i)3-s + (3.46 + 5.99i)4-s + (30.3 + 17.5i)5-s + (9.10 − 42.1i)6-s + (48.8 − 3.13i)7-s + 43.4i·8-s + (−65.8 + 47.1i)9-s + (−83.7 − 145. i)10-s + (−21.3 + 12.3i)11-s + (−41.8 + 46.1i)12-s − 105.·13-s + (−210. − 104. i)14-s + (−66.5 + 307. i)15-s + (159. − 276. i)16-s + (365. − 211. i)17-s + ⋯
L(s)  = 1  + (−1.03 − 0.598i)2-s + (0.305 + 0.952i)3-s + (0.216 + 0.374i)4-s + (1.21 + 0.700i)5-s + (0.252 − 1.16i)6-s + (0.997 − 0.0639i)7-s + 0.679i·8-s + (−0.813 + 0.582i)9-s + (−0.837 − 1.45i)10-s + (−0.176 + 0.102i)11-s + (−0.290 + 0.320i)12-s − 0.625·13-s + (−1.07 − 0.530i)14-s + (−0.295 + 1.36i)15-s + (0.622 − 1.07i)16-s + (1.26 − 0.731i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(2.17076\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :2),\ 0.905 - 0.424i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.961172 + 0.213940i\)
\(L(\frac12)\) \(\approx\) \(0.961172 + 0.213940i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.75 - 8.56i)T \)
7 \( 1 + (-48.8 + 3.13i)T \)
good2 \( 1 + (4.14 + 2.39i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + (-30.3 - 17.5i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (21.3 - 12.3i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 105.T + 2.85e4T^{2} \)
17 \( 1 + (-365. + 211. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (13.4 - 23.3i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (580. + 334. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 332. iT - 7.07e5T^{2} \)
31 \( 1 + (-35.0 - 60.6i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-601. + 1.04e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 1.61e3iT - 2.82e6T^{2} \)
43 \( 1 + 309.T + 3.41e6T^{2} \)
47 \( 1 + (2.98e3 + 1.72e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.63e3 + 1.52e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-384. + 222. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.98e3 + 5.16e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (1.67e3 + 2.89e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 6.99e3iT - 2.54e7T^{2} \)
73 \( 1 + (-681. - 1.18e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.04e3 - 8.74e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 461. iT - 4.74e7T^{2} \)
89 \( 1 + (-2.11e3 - 1.22e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 6.03e3T + 8.85e7T^{2} \)
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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

−17.71318017793554203768715355237, −16.65770265124514696486017493199, −14.63168168361855793516140225246, −14.11893299370946289190761281541, −11.49976124683728700658577960711, −10.24272779914391569892486779633, −9.694038859666028970833320458208, −8.091969076984559354503568898034, −5.31586157561232279469080273571, −2.33484256665381316649402326706, 1.48177338138975177424949868852, 5.79678249419829060265373991746, 7.62152805585484085816788474224, 8.639720123364365656807015400955, 9.938996212845074142344664457571, 12.22531903146698309256626033563, 13.46428291057568009811593237749, 14.70736602757402502588135331064, 16.67035563139943523216793467220, 17.50637658532334450827485175805

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