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Properties

Label 2-21-21.17-c19-0-34
Degree $2$
Conductor $21$
Sign $0.941 + 0.337i$
Analytic cond. $48.0515$
Root an. cond. $6.93191$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.95e4 + 1.70e4i)3-s + (−2.62e5 + 4.54e5i)4-s + (4.84e7 − 9.51e7i)7-s + (5.81e8 + 1.00e9i)9-s + (−1.54e10 + 8.93e9i)12-s − 7.60e10i·13-s + (−1.37e11 − 2.38e11i)16-s + (1.02e12 − 5.91e11i)19-s + (3.05e12 − 1.98e12i)21-s + (9.53e12 − 1.65e13i)25-s + (0.00195 + 3.96e13i)27-s + (3.05e13 + 4.69e13i)28-s + (−2.39e14 − 1.38e14i)31-s − 6.09e14·36-s + (7.90e14 + 1.36e15i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.453 − 0.891i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s − 1.98i·13-s + (−0.499 − 0.866i)16-s + (0.727 − 0.420i)19-s + (0.838 − 0.545i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.545 + 0.838i)28-s + (−1.62 − 0.939i)31-s − 36-s + (0.999 + 1.73i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(48.0515\)
Root analytic conductor: \(6.93191\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :19/2),\ 0.941 + 0.337i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.505430003\)
\(L(\frac12)\) \(\approx\) \(2.505430003\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.95e4 - 1.70e4i)T \)
7 \( 1 + (-4.84e7 + 9.51e7i)T \)
good2 \( 1 + (2.62e5 - 4.54e5i)T^{2} \)
5 \( 1 + (-9.53e12 + 1.65e13i)T^{2} \)
11 \( 1 + (3.05e19 + 5.29e19i)T^{2} \)
13 \( 1 + 7.60e10iT - 1.46e21T^{2} \)
17 \( 1 + (-1.19e23 - 2.07e23i)T^{2} \)
19 \( 1 + (-1.02e12 + 5.91e11i)T + (9.89e23 - 1.71e24i)T^{2} \)
23 \( 1 + (3.73e25 - 6.46e25i)T^{2} \)
29 \( 1 - 6.10e27T^{2} \)
31 \( 1 + (2.39e14 + 1.38e14i)T + (1.08e28 + 1.87e28i)T^{2} \)
37 \( 1 + (-7.90e14 - 1.36e15i)T + (-3.12e29 + 5.41e29i)T^{2} \)
41 \( 1 + 4.39e30T^{2} \)
43 \( 1 - 6.01e15T + 1.08e31T^{2} \)
47 \( 1 + (-2.94e31 + 5.09e31i)T^{2} \)
53 \( 1 + (2.88e32 + 4.99e32i)T^{2} \)
59 \( 1 + (-2.21e33 - 3.83e33i)T^{2} \)
61 \( 1 + (-1.01e17 + 5.83e16i)T + (4.17e33 - 7.22e33i)T^{2} \)
67 \( 1 + (-2.20e17 + 3.82e17i)T + (-2.47e34 - 4.29e34i)T^{2} \)
71 \( 1 - 1.49e35T^{2} \)
73 \( 1 + (1.43e17 + 8.27e16i)T + (1.26e35 + 2.19e35i)T^{2} \)
79 \( 1 + (1.31e17 + 2.27e17i)T + (-5.67e35 + 9.82e35i)T^{2} \)
83 \( 1 + 2.90e36T^{2} \)
89 \( 1 + (-5.46e36 + 9.46e36i)T^{2} \)
97 \( 1 - 1.15e19iT - 5.60e37T^{2} \)
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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

−13.64612183099239223210685742333, −12.78414203779864028188276129199, −10.89251528305971152572764262783, −9.642439526416713733468941961207, −8.201421185858428547167459474895, −7.54060035481372786026296431734, −4.99621583612341112379524321898, −3.78894834793450174241616401295, −2.75231405181806532155629499226, −0.67109127058146114767755007230, 1.26911196060362356602565407853, 2.19353664885276320573602748104, 4.07941032490452674832954258172, 5.63023582551122440147807062809, 7.13682848223445912008143925225, 8.861300666120112080790741056391, 9.400428685343997025739366777072, 11.31751567418832083638585903401, 12.73358025340843565084821453713, 14.21588995834844244517892802251

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