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Properties

Label 2-21-21.2-c4-0-2
Degree $2$
Conductor $21$
Sign $0.372 - 0.928i$
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  + (4.46 + 2.58i)2-s + (−6.70 + 5.99i)3-s + (5.31 + 9.20i)4-s + (31.7 + 18.3i)5-s + (−45.4 + 9.49i)6-s + (−18.9 − 45.2i)7-s − 27.6i·8-s + (9.04 − 80.4i)9-s + (94.5 + 163. i)10-s + (−32.7 + 18.9i)11-s + (−90.9 − 29.8i)12-s + 127.·13-s + (32.0 − 250. i)14-s + (−322. + 67.4i)15-s + (156. − 271. i)16-s + (−248. + 143. i)17-s + ⋯
L(s)  = 1  + (1.11 + 0.645i)2-s + (−0.745 + 0.666i)3-s + (0.332 + 0.575i)4-s + (1.26 + 0.732i)5-s + (−1.26 + 0.263i)6-s + (−0.386 − 0.922i)7-s − 0.432i·8-s + (0.111 − 0.993i)9-s + (0.945 + 1.63i)10-s + (−0.270 + 0.156i)11-s + (−0.631 − 0.207i)12-s + 0.752·13-s + (0.163 − 1.27i)14-s + (−1.43 + 0.299i)15-s + (0.611 − 1.05i)16-s + (−0.859 + 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.372 - 0.928i$
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: \((2,\ 21,\ (\ :2),\ 0.372 - 0.928i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.56399 + 1.05762i\)
\(L(\frac12)\) \(\approx\) \(1.56399 + 1.05762i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.70 - 5.99i)T \)
7 \( 1 + (18.9 + 45.2i)T \)
good2 \( 1 + (-4.46 - 2.58i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + (-31.7 - 18.3i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (32.7 - 18.9i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 127.T + 2.85e4T^{2} \)
17 \( 1 + (248. - 143. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (161. - 280. i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (303. + 175. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 210. iT - 7.07e5T^{2} \)
31 \( 1 + (658. + 1.13e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (497. - 861. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 288. iT - 2.82e6T^{2} \)
43 \( 1 - 1.81e3T + 3.41e6T^{2} \)
47 \( 1 + (-91.1 - 52.6i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.85e3 + 1.64e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (4.70e3 - 2.71e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.90e3 + 3.30e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-4.33e3 - 7.50e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 3.28e3iT - 2.54e7T^{2} \)
73 \( 1 + (-2.76e3 - 4.79e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-384. + 665. i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 3.70e3iT - 4.74e7T^{2} \)
89 \( 1 + (-4.32e3 - 2.49e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 1.96e3T + 8.85e7T^{2} \)
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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

−17.28483198107953749890370986693, −16.21857032745641580620920296525, −14.96998536227305343664213684594, −13.88178808741900400630910545974, −12.87843745855314554344168598319, −10.75651102151874453081187141814, −9.837317066102287280718745931850, −6.66183661460829687980957385968, −5.82372260085769629693896945292, −4.00552268290051356058457023530, 2.14908222858195243301578847850, 5.12848509328607877220537272153, 6.11886067947336070523637133918, 8.898176684140438662874025458365, 10.90950629193390040055288856365, 12.26600367312821236885499496381, 13.11403673086840248383671974393, 13.82735021957639695308805226365, 15.88823059409624162069020363835, 17.37273016954652365109808795885

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