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Properties

Label 2-21-21.20-c15-0-8
Degree $2$
Conductor $21$
Sign $-0.958 - 0.285i$
Analytic cond. $29.9656$
Root an. cond. $5.47408$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.78e3i·3-s + 3.27e4·4-s + (−6.22e5 + 2.08e6i)7-s − 1.43e7·9-s + 1.24e8i·12-s + 2.15e8i·13-s + 1.07e9·16-s − 1.19e9i·19-s + (−7.90e9 − 2.35e9i)21-s − 3.05e10·25-s − 5.43e10i·27-s + (−2.03e10 + 6.84e10i)28-s + 2.19e11i·31-s − 4.70e11·36-s − 1.09e12·37-s + ⋯
L(s)  = 1  + 1.00i·3-s + 4-s + (−0.285 + 0.958i)7-s − 1.00·9-s + 1.00i·12-s + 0.953i·13-s + 16-s − 0.306i·19-s + (−0.958 − 0.285i)21-s − 25-s − 1.00i·27-s + (−0.285 + 0.958i)28-s + 1.43i·31-s − 1.00·36-s − 1.88·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(29.9656\)
Root analytic conductor: \(5.47408\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :15/2),\ -0.958 - 0.285i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.658060992\)
\(L(\frac12)\) \(\approx\) \(1.658060992\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3.78e3iT \)
7 \( 1 + (6.22e5 - 2.08e6i)T \)
good2 \( 1 - 3.27e4T^{2} \)
5 \( 1 + 3.05e10T^{2} \)
11 \( 1 - 4.17e15T^{2} \)
13 \( 1 - 2.15e8iT - 5.11e16T^{2} \)
17 \( 1 + 2.86e18T^{2} \)
19 \( 1 + 1.19e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.66e20T^{2} \)
29 \( 1 - 8.62e21T^{2} \)
31 \( 1 - 2.19e11iT - 2.34e22T^{2} \)
37 \( 1 + 1.09e12T + 3.33e23T^{2} \)
41 \( 1 + 1.55e24T^{2} \)
43 \( 1 + 1.44e12T + 3.17e24T^{2} \)
47 \( 1 + 1.20e25T^{2} \)
53 \( 1 - 7.31e25T^{2} \)
59 \( 1 + 3.65e26T^{2} \)
61 \( 1 + 2.81e13iT - 6.02e26T^{2} \)
67 \( 1 - 9.90e13T + 2.46e27T^{2} \)
71 \( 1 - 5.87e27T^{2} \)
73 \( 1 - 1.88e14iT - 8.90e27T^{2} \)
79 \( 1 + 8.86e13T + 2.91e28T^{2} \)
83 \( 1 + 6.11e28T^{2} \)
89 \( 1 + 1.74e29T^{2} \)
97 \( 1 - 1.20e15iT - 6.33e29T^{2} \)
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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

−15.49324509119703693504009919440, −14.24082859765475427127382918343, −12.18309682557366742813087522021, −11.24357202339979074206706482467, −9.895183044066066667924107343752, −8.594137534394891621487304573129, −6.63712617479463320711074826739, −5.28425699911513668261171264844, −3.42968457279867998317305634211, −2.07261344498227344383184156871, 0.46919766004656336527176456339, 1.84438923898225316045784140169, 3.33454189204525733152853180207, 5.84155133974696674709532997182, 7.06305225098507092317833600335, 8.005847599001367050501114729258, 10.22921667753242064975918382585, 11.47215887844121575066050454796, 12.68591442736919373644606412592, 13.82914797863283444878026347711

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