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Properties

Label 2-21-21.2-c2-0-0
Degree $2$
Conductor $21$
Sign $0.978 - 0.205i$
Analytic cond. $0.572208$
Root an. cond. $0.756444$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s + (−2 − 3.46i)4-s + (−6.5 − 2.59i)7-s + (−4.5 + 7.79i)9-s + (6 − 10.3i)12-s + 23·13-s + (−7.99 + 13.8i)16-s + (−5.5 + 9.52i)19-s + (−3 − 20.7i)21-s + (−12.5 − 21.6i)25-s − 27·27-s + (4 + 27.7i)28-s + (6.5 + 11.2i)31-s + 36·36-s + (36.5 − 63.2i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.928 − 0.371i)7-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)12-s + 1.76·13-s + (−0.499 + 0.866i)16-s + (−0.289 + 0.501i)19-s + (−0.142 − 0.989i)21-s + (−0.5 − 0.866i)25-s − 27-s + (0.142 + 0.989i)28-s + (0.209 + 0.363i)31-s + 36-s + (0.986 − 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.978 - 0.205i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.978 - 0.205i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.877856 + 0.0910609i\)
\(L(\frac12)\) \(\approx\) \(0.877856 + 0.0910609i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (6.5 + 2.59i)T \)
good2 \( 1 + (2 + 3.46i)T^{2} \)
5 \( 1 + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 - 23T + 169T^{2} \)
17 \( 1 + (144.5 - 250. i)T^{2} \)
19 \( 1 + (5.5 - 9.52i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (-6.5 - 11.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-36.5 + 63.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37 - 64.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-48.5 - 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 2T + 9.40e3T^{2} \)
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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

−18.22591168197302180246445512932, −16.41480989137551148618680286450, −15.53367582927521313680246387256, −14.19262124482623320311386052803, −13.25824231990411862652613133590, −10.85867357147380919384377244288, −9.853380481351000633855114089440, −8.616907772579954865039179977931, −6.00577066649538924854715699344, −3.95940027798559613597211892625, 3.36578704399175758378478904088, 6.45358831043530219069890543980, 8.157996158947256526575565289898, 9.252588891088115612641962602212, 11.67230751947332469346862091920, 13.05608419011204828296859916744, 13.54581719826265743074080423457, 15.42721410848876837977762970691, 16.84208441965893813213973103800, 18.18469278944139048887368297192

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