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Properties

Label 2-21-21.2-c40-0-27
Degree $2$
Conductor $21$
Sign $0.331 + 0.943i$
Analytic cond. $212.818$
Root an. cond. $14.5883$
Motivic weight $40$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.74e9 − 3.01e9i)3-s + (−5.49e11 − 9.52e11i)4-s + (1.96e16 − 7.73e16i)7-s + (−6.07e18 + 1.05e19i)9-s + (−1.91e21 + 3.32e21i)12-s − 3.65e22·13-s + (−6.04e23 + 1.04e24i)16-s + (−3.44e25 + 5.97e25i)19-s + (−2.67e26 + 7.54e25i)21-s + (−4.54e27 − 7.87e27i)25-s + (4.23e28 − 4.39e12i)27-s + (−8.44e28 + 2.37e28i)28-s + (3.14e29 + 5.45e29i)31-s + (1.33e31 − 1.12e15i)36-s + (2.28e31 − 3.95e31i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.246 − 0.969i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s − 1.92·13-s + (−0.499 + 0.866i)16-s + (−0.917 + 1.58i)19-s + (−0.962 + 0.271i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−0.962 + 0.271i)28-s + (0.468 + 0.811i)31-s + 0.999·36-s + (0.988 − 1.71i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(212.818\)
Root analytic conductor: \(14.5883\)
Motivic weight: \(40\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :20),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(0.5066942827\)
\(L(\frac12)\) \(\approx\) \(0.5066942827\)
\(L(21)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.74e9 + 3.01e9i)T \)
7 \( 1 + (-1.96e16 + 7.73e16i)T \)
good2 \( 1 + (5.49e11 + 9.52e11i)T^{2} \)
5 \( 1 + (4.54e27 + 7.87e27i)T^{2} \)
11 \( 1 + (2.26e41 - 3.91e41i)T^{2} \)
13 \( 1 + 3.65e22T + 3.61e44T^{2} \)
17 \( 1 + (8.25e48 - 1.43e49i)T^{2} \)
19 \( 1 + (3.44e25 - 5.97e25i)T + (-7.06e50 - 1.22e51i)T^{2} \)
23 \( 1 + (1.47e54 + 2.55e54i)T^{2} \)
29 \( 1 - 3.13e58T^{2} \)
31 \( 1 + (-3.14e29 - 5.45e29i)T + (-2.25e59 + 3.90e59i)T^{2} \)
37 \( 1 + (-2.28e31 + 3.95e31i)T + (-2.67e62 - 4.63e62i)T^{2} \)
41 \( 1 - 3.24e64T^{2} \)
43 \( 1 + 9.32e32T + 2.18e65T^{2} \)
47 \( 1 + (3.82e66 + 6.62e66i)T^{2} \)
53 \( 1 + (4.67e68 - 8.10e68i)T^{2} \)
59 \( 1 + (3.41e70 - 5.91e70i)T^{2} \)
61 \( 1 + (4.54e35 - 7.86e35i)T + (-1.29e71 - 2.24e71i)T^{2} \)
67 \( 1 + (-1.20e36 - 2.09e36i)T + (-5.52e72 + 9.56e72i)T^{2} \)
71 \( 1 - 1.12e74T^{2} \)
73 \( 1 + (-7.12e36 - 1.23e37i)T + (-1.70e74 + 2.95e74i)T^{2} \)
79 \( 1 + (1.58e37 - 2.73e37i)T + (-4.01e75 - 6.96e75i)T^{2} \)
83 \( 1 - 5.79e76T^{2} \)
89 \( 1 + (4.72e77 + 8.18e77i)T^{2} \)
97 \( 1 - 1.06e40T + 2.95e79T^{2} \)
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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

−10.56576771991268565740923633870, −9.925209436365992630762187565304, −8.247166687802911032388498144196, −7.24310732322409152547364960980, −6.18404207871171335409812123101, −5.10316967526853517898843160731, −4.19047607726869185562613734658, −2.28802849856753639709373396592, −1.40682969018832242539461663979, −0.35094605161172732436381225610, 0.25862881723275194764238812164, 2.32591173483457708905891533144, 3.18725160151216245245618082681, 4.66558057168868896953643085496, 4.98752868265940202214269557792, 6.55182184330857643523435020924, 7.952013836118456364500283061504, 9.100628285859780441724535281976, 9.795930727703428107175642101971, 11.41228011742615635419879500045

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