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Properties

Label 2-21-21.20-c29-0-64
Degree $2$
Conductor $21$
Sign $-0.169 + 0.985i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.64e3i·2-s + (2.25e6 + 7.97e6i)3-s + 5.34e8·4-s + 1.37e10·5-s + (−1.31e10 + 3.71e9i)6-s + (−1.61e12 − 7.74e11i)7-s + 1.76e12i·8-s + (−5.84e13 + 3.60e13i)9-s + 2.25e13i·10-s − 1.65e15i·11-s + (1.20e15 + 4.25e15i)12-s + 7.11e15i·13-s + (1.27e15 − 2.66e15i)14-s + (3.09e16 + 1.09e17i)15-s + 2.83e17·16-s − 1.14e18·17-s + ⋯
L(s)  = 1  + 0.0710i·2-s + (0.272 + 0.962i)3-s + 0.994·4-s + 1.00·5-s + (−0.0683 + 0.0193i)6-s + (−0.902 − 0.431i)7-s + 0.141i·8-s + (−0.851 + 0.524i)9-s + 0.0713i·10-s − 1.31i·11-s + (0.271 + 0.957i)12-s + 0.501i·13-s + (0.0306 − 0.0640i)14-s + (0.273 + 0.966i)15-s + 0.984·16-s − 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.169 + 0.985i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.169 + 0.985i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.108386889\)
\(L(\frac12)\) \(\approx\) \(1.108386889\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.25e6 - 7.97e6i)T \)
7 \( 1 + (1.61e12 + 7.74e11i)T \)
good2 \( 1 - 1.64e3iT - 5.36e8T^{2} \)
5 \( 1 - 1.37e10T + 1.86e20T^{2} \)
11 \( 1 + 1.65e15iT - 1.58e30T^{2} \)
13 \( 1 - 7.11e15iT - 2.01e32T^{2} \)
17 \( 1 + 1.14e18T + 4.81e35T^{2} \)
19 \( 1 - 4.19e18iT - 1.21e37T^{2} \)
23 \( 1 + 6.01e19iT - 3.09e39T^{2} \)
29 \( 1 + 2.75e21iT - 2.56e42T^{2} \)
31 \( 1 + 2.77e21iT - 1.77e43T^{2} \)
37 \( 1 + 2.45e22T + 3.00e45T^{2} \)
41 \( 1 - 2.01e23T + 5.89e46T^{2} \)
43 \( 1 + 7.28e23T + 2.34e47T^{2} \)
47 \( 1 + 2.16e24T + 3.09e48T^{2} \)
53 \( 1 - 1.10e25iT - 1.00e50T^{2} \)
59 \( 1 - 2.77e25T + 2.26e51T^{2} \)
61 \( 1 + 9.03e25iT - 5.95e51T^{2} \)
67 \( 1 + 2.34e25T + 9.04e52T^{2} \)
71 \( 1 + 4.44e26iT - 4.85e53T^{2} \)
73 \( 1 - 7.29e25iT - 1.08e54T^{2} \)
79 \( 1 + 4.11e27T + 1.07e55T^{2} \)
83 \( 1 + 7.11e27T + 4.50e55T^{2} \)
89 \( 1 + 1.23e28T + 3.40e56T^{2} \)
97 \( 1 - 2.43e28iT - 4.13e57T^{2} \)
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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

−11.46166665818701802310391607315, −10.47658531964703200588975469072, −9.591829731924725026999553543930, −8.295519331481854857786780824404, −6.45282974872739221024113866637, −5.87667651821187057498040163196, −4.09888840091381232685735218033, −2.91283102153355328417771346838, −1.97140915332672975170574277906, −0.17622199591706676004273858572, 1.50867867311301992615715294438, 2.22958578645548043744701240432, 3.08984629673409662904591451752, 5.38503753528372772598675835042, 6.64218230301870588477511239070, 7.06619293107640718646506615921, 8.866479980266257740014330486309, 9.984635466155871723129223013253, 11.44476050901784017030492683963, 12.71741448989665849338045961197

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