Properties

Label 2-21-3.2-c6-0-1
Degree $2$
Conductor $21$
Sign $-0.683 + 0.730i$
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  + 12.1i·2-s + (−18.4 + 19.7i)3-s − 84.1·4-s + 4.50i·5-s + (−239. − 224. i)6-s + 129.·7-s − 245. i·8-s + (−48.3 − 727. i)9-s − 54.8·10-s + 332. i·11-s + (1.55e3 − 1.65e3i)12-s − 3.03e3·13-s + 1.57e3i·14-s + (−88.8 − 83.1i)15-s − 2.39e3·16-s + 8.74e3i·17-s + ⋯
L(s)  = 1  + 1.52i·2-s + (−0.683 + 0.730i)3-s − 1.31·4-s + 0.0360i·5-s + (−1.11 − 1.03i)6-s + 0.377·7-s − 0.479i·8-s + (−0.0662 − 0.997i)9-s − 0.0548·10-s + 0.249i·11-s + (0.898 − 0.960i)12-s − 1.38·13-s + 0.575i·14-s + (−0.0263 − 0.0246i)15-s − 0.585·16-s + 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.683 + 0.730i$
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: \((2,\ 21,\ (\ :3),\ -0.683 + 0.730i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.345540 - 0.796579i\)
\(L(\frac12)\) \(\approx\) \(0.345540 - 0.796579i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (18.4 - 19.7i)T \)
7 \( 1 - 129.T \)
good2 \( 1 - 12.1iT - 64T^{2} \)
5 \( 1 - 4.50iT - 1.56e4T^{2} \)
11 \( 1 - 332. iT - 1.77e6T^{2} \)
13 \( 1 + 3.03e3T + 4.82e6T^{2} \)
17 \( 1 - 8.74e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.52e3T + 4.70e7T^{2} \)
23 \( 1 + 305. iT - 1.48e8T^{2} \)
29 \( 1 - 1.59e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.55e4T + 8.87e8T^{2} \)
37 \( 1 - 6.19e4T + 2.56e9T^{2} \)
41 \( 1 - 2.75e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.24e5T + 6.32e9T^{2} \)
47 \( 1 + 2.45e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.57e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.58e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.15e5T + 5.15e10T^{2} \)
67 \( 1 + 3.28e5T + 9.04e10T^{2} \)
71 \( 1 + 3.84e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.40e5T + 1.51e11T^{2} \)
79 \( 1 - 4.40e5T + 2.43e11T^{2} \)
83 \( 1 + 1.00e5iT - 3.26e11T^{2} \)
89 \( 1 + 9.61e5iT - 4.96e11T^{2} \)
97 \( 1 - 9.73e5T + 8.32e11T^{2} \)
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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

−17.14868752418632570438686352745, −16.48635379530151892435478930595, −15.07576510392426160328613424905, −14.59762349024471527576682638337, −12.49699252781275290308790938416, −10.74559116156133546126421396122, −9.121156056603143120967478279267, −7.43379741822243395588459017405, −5.90122800610158876267459602792, −4.60098212351708496930235408665, 0.59708101761326252455495282298, 2.44922258492772732693019279014, 4.96771715157316624203870195215, 7.32023069555729014993812100560, 9.511664244507354622218254438873, 11.01027650271654721574494093637, 11.87529495814664166295363058872, 12.87120142845811711456894338796, 14.16996208224788468337306882437, 16.38972393409448959305300798895

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