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Properties

Label 2-21-3.2-c6-0-8
Degree $2$
Conductor $21$
Sign $0.498 + 0.867i$
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  + 11.9i·2-s + (−13.4 − 23.4i)3-s − 77.8·4-s − 201. i·5-s + (278. − 160. i)6-s − 129.·7-s − 164. i·8-s + (−367. + 629. i)9-s + 2.39e3·10-s − 2.45e3i·11-s + (1.04e3 + 1.82e3i)12-s − 238.·13-s − 1.54e3i·14-s + (−4.71e3 + 2.70e3i)15-s − 3.02e3·16-s − 2.17e3i·17-s + ⋯
L(s)  = 1  + 1.48i·2-s + (−0.498 − 0.867i)3-s − 1.21·4-s − 1.61i·5-s + (1.29 − 0.741i)6-s − 0.377·7-s − 0.321i·8-s + (−0.503 + 0.863i)9-s + 2.39·10-s − 1.84i·11-s + (0.605 + 1.05i)12-s − 0.108·13-s − 0.562i·14-s + (−1.39 + 0.802i)15-s − 0.737·16-s − 0.441i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.498 + 0.867i$
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: \((2,\ 21,\ (\ :3),\ 0.498 + 0.867i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.741172 - 0.428984i\)
\(L(\frac12)\) \(\approx\) \(0.741172 - 0.428984i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.4 + 23.4i)T \)
7 \( 1 + 129.T \)
good2 \( 1 - 11.9iT - 64T^{2} \)
5 \( 1 + 201. iT - 1.56e4T^{2} \)
11 \( 1 + 2.45e3iT - 1.77e6T^{2} \)
13 \( 1 + 238.T + 4.82e6T^{2} \)
17 \( 1 + 2.17e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.03e3T + 4.70e7T^{2} \)
23 \( 1 - 7.12e3iT - 1.48e8T^{2} \)
29 \( 1 - 3.39e3iT - 5.94e8T^{2} \)
31 \( 1 - 2.42e4T + 8.87e8T^{2} \)
37 \( 1 - 4.29e4T + 2.56e9T^{2} \)
41 \( 1 - 1.19e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.05e4T + 6.32e9T^{2} \)
47 \( 1 + 1.22e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.28e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.38e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.88e4T + 5.15e10T^{2} \)
67 \( 1 - 5.60e5T + 9.04e10T^{2} \)
71 \( 1 + 5.06e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.03e5T + 1.51e11T^{2} \)
79 \( 1 + 3.08e5T + 2.43e11T^{2} \)
83 \( 1 + 1.07e6iT - 3.26e11T^{2} \)
89 \( 1 - 4.65e5iT - 4.96e11T^{2} \)
97 \( 1 - 7.00e5T + 8.32e11T^{2} \)
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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

−16.57864382171532715470134220897, −15.92538517246323436665945312391, −13.87791022541791961021309726653, −13.10149663670301170006109168552, −11.60189033481145095149192880346, −8.881320983603828991256454143709, −7.942455502671297554683023933163, −6.21646528576664739193444100394, −5.17751341206966365382958125441, −0.57204046252275183881874679296, 2.62505627958476827898364828270, 4.17058359341817361858497612376, 6.72232078358194690652079989617, 9.755440944433188247846410476069, 10.33174375209756811569464829827, 11.41038257982724240399308293028, 12.59553930550378895551796836790, 14.53323534223346407260793697105, 15.52007346910807240286823453588, 17.49915474784570536170231203250

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