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Properties

Label 2-21-21.11-c34-0-41
Degree $2$
Conductor $21$
Sign $0.719 + 0.694i$
Analytic cond. $153.773$
Root an. cond. $12.4005$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.45e7 − 1.11e8i)3-s + (−8.58e9 + 1.48e10i)4-s + (−2.28e14 − 4.18e13i)7-s + (−8.33e15 − 1.44e16i)9-s + (1.10e18 + 1.92e18i)12-s − 6.62e18·13-s + (−1.47e20 − 2.55e20i)16-s + (5.26e21 + 9.12e21i)19-s + (−1.94e22 + 2.28e22i)21-s + (−2.91e23 + 5.04e23i)25-s − 2.15e24·27-s + (2.58e24 − 3.04e24i)28-s + (−9.79e24 + 1.69e25i)31-s + 2.86e26·36-s + (−4.29e26 − 7.44e26i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.983 − 0.180i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s − 0.766·13-s + (−0.499 − 0.866i)16-s + (0.961 + 1.66i)19-s + (−0.647 + 0.761i)21-s + (−0.5 + 0.866i)25-s − 27-s + (0.647 − 0.761i)28-s + (−0.434 + 0.752i)31-s + 0.999·36-s + (−0.941 − 1.63i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(153.773\)
Root analytic conductor: \(12.4005\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :17),\ 0.719 + 0.694i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.126139324\)
\(L(\frac12)\) \(\approx\) \(1.126139324\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.45e7 + 1.11e8i)T \)
7 \( 1 + (2.28e14 + 4.18e13i)T \)
good2 \( 1 + (8.58e9 - 1.48e10i)T^{2} \)
5 \( 1 + (2.91e23 - 5.04e23i)T^{2} \)
11 \( 1 + (1.27e35 + 2.21e35i)T^{2} \)
13 \( 1 + 6.62e18T + 7.48e37T^{2} \)
17 \( 1 + (3.42e41 + 5.92e41i)T^{2} \)
19 \( 1 + (-5.26e21 - 9.12e21i)T + (-1.50e43 + 2.60e43i)T^{2} \)
23 \( 1 + (9.94e45 - 1.72e46i)T^{2} \)
29 \( 1 - 5.26e49T^{2} \)
31 \( 1 + (9.79e24 - 1.69e25i)T + (-2.54e50 - 4.40e50i)T^{2} \)
37 \( 1 + (4.29e26 + 7.44e26i)T + (-1.04e53 + 1.80e53i)T^{2} \)
41 \( 1 - 6.83e54T^{2} \)
43 \( 1 + 8.73e27T + 3.45e55T^{2} \)
47 \( 1 + (3.55e56 - 6.14e56i)T^{2} \)
53 \( 1 + (2.11e58 + 3.65e58i)T^{2} \)
59 \( 1 + (8.08e59 + 1.40e60i)T^{2} \)
61 \( 1 + (-2.23e30 - 3.86e30i)T + (-2.51e60 + 4.35e60i)T^{2} \)
67 \( 1 + (-1.10e31 + 1.90e31i)T + (-6.10e61 - 1.05e62i)T^{2} \)
71 \( 1 - 8.76e62T^{2} \)
73 \( 1 + (1.00e31 - 1.73e31i)T + (-1.12e63 - 1.95e63i)T^{2} \)
79 \( 1 + (1.68e32 + 2.91e32i)T + (-1.65e64 + 2.86e64i)T^{2} \)
83 \( 1 - 1.77e65T^{2} \)
89 \( 1 + (9.51e65 - 1.64e66i)T^{2} \)
97 \( 1 - 2.07e33T + 3.55e67T^{2} \)
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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

−11.92892327972585138115068638132, −9.900583166499109113097063435412, −8.928591064303356451406515489443, −7.72139976650317004555121011774, −7.00918517389730965478980366061, −5.54450779468948681639259078644, −3.73863398151794588108003426572, −3.13697337204119849076936390629, −1.80771587328188239866442519094, −0.36810713432593345669034082371, 0.53133401046819750710553415806, 2.22978357247250504686401693917, 3.30481785399499313485791623070, 4.59353984637695093275728395201, 5.44134045562003713435175293875, 6.82503761052307169169953641636, 8.530978379986617246910474480663, 9.617214174830801738127082890776, 10.02988359020886324063871312727, 11.45767033506081402323746126972

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