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Properties

Label 2-21-3.2-c6-0-2
Degree $2$
Conductor $21$
Sign $-0.999 + 0.0199i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.2i·2-s + (26.9 − 0.538i)3-s − 168.·4-s + 118. i·5-s + (8.22 + 412. i)6-s − 129.·7-s − 1.60e3i·8-s + (728. − 29.0i)9-s − 1.80e3·10-s + 356. i·11-s + (−4.56e3 + 91.0i)12-s + 2.04e3·13-s − 1.97e3i·14-s + (63.8 + 3.19e3i)15-s + 1.36e4·16-s + 4.81e3i·17-s + ⋯
L(s)  = 1  + 1.90i·2-s + (0.999 − 0.0199i)3-s − 2.64·4-s + 0.947i·5-s + (0.0380 + 1.90i)6-s − 0.377·7-s − 3.12i·8-s + (0.999 − 0.0399i)9-s − 1.80·10-s + 0.267i·11-s + (−2.63 + 0.0526i)12-s + 0.929·13-s − 0.721i·14-s + (0.0189 + 0.947i)15-s + 3.32·16-s + 0.979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0199i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.999 + 0.0199i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.999 + 0.0199i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0166804 - 1.67171i\)
\(L(\frac12)\) \(\approx\) \(0.0166804 - 1.67171i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-26.9 + 0.538i)T \)
7 \( 1 + 129.T \)
good2 \( 1 - 15.2iT - 64T^{2} \)
5 \( 1 - 118. iT - 1.56e4T^{2} \)
11 \( 1 - 356. iT - 1.77e6T^{2} \)
13 \( 1 - 2.04e3T + 4.82e6T^{2} \)
17 \( 1 - 4.81e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.51e3T + 4.70e7T^{2} \)
23 \( 1 - 1.38e4iT - 1.48e8T^{2} \)
29 \( 1 + 4.23e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.55e4T + 8.87e8T^{2} \)
37 \( 1 + 7.62e3T + 2.56e9T^{2} \)
41 \( 1 + 5.54e4iT - 4.75e9T^{2} \)
43 \( 1 - 6.02e4T + 6.32e9T^{2} \)
47 \( 1 + 1.29e5iT - 1.07e10T^{2} \)
53 \( 1 + 6.09e4iT - 2.21e10T^{2} \)
59 \( 1 + 4.03e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.18e5T + 5.15e10T^{2} \)
67 \( 1 + 1.33e5T + 9.04e10T^{2} \)
71 \( 1 - 1.95e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.44e5T + 1.51e11T^{2} \)
79 \( 1 + 8.94e5T + 2.43e11T^{2} \)
83 \( 1 - 2.87e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.99e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.17e6T + 8.32e11T^{2} \)
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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

−17.30123217201195487578072269220, −15.72365203468963279540315473210, −15.13413558489317571895211201498, −14.04372724183913821979196554220, −13.14058727972423517922507314774, −10.02426809907462105583282085456, −8.632903152377926414613178354068, −7.41363276705162195615653370507, −6.21210053181659855246638054320, −3.84382803302262106565075646599, 1.11835804180406885395044860654, 2.99262467366116793143473840113, 4.54495670151747176774254494294, 8.540195024198979294090051593147, 9.301472600743062535927129080461, 10.71353376258705668171790451764, 12.37383673655709179274421438985, 13.18281093210418841374466517097, 14.21240929626065416019255152689, 16.27729334730637207202355048276

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