Properties

Label 2-21-7.4-c29-0-25
Degree $2$
Conductor $21$
Sign $0.762 - 0.647i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.30e4 + 3.98e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (−7.90e8 + 1.36e9i)4-s + (1.61e9 + 2.79e9i)5-s − 2.20e11·6-s + (−4.94e11 + 1.72e12i)7-s + (−4.80e13 + 0.00390i)8-s + (−1.14e13 − 1.98e13i)9-s + (−7.43e13 + 1.28e14i)10-s + (3.88e13 − 6.73e13i)11-s + (−3.78e15 − 6.54e15i)12-s + 1.49e16·13-s + (−8.01e16 + 1.99e16i)14-s − 1.54e16·15-s + (−6.81e17 − 1.18e18i)16-s + (4.88e15 − 8.46e15i)17-s + ⋯
L(s)  = 1  + (0.993 + 1.72i)2-s + (−0.288 + 0.499i)3-s + (−1.47 + 2.55i)4-s + (0.118 + 0.204i)5-s − 1.14·6-s + (−0.275 + 0.961i)7-s − 3.86·8-s + (−0.166 − 0.288i)9-s + (−0.234 + 0.406i)10-s + (0.0308 − 0.0534i)11-s + (−0.850 − 1.47i)12-s + 1.04·13-s + (−1.92 + 0.480i)14-s − 0.136·15-s + (−2.36 − 4.09i)16-s + (0.00704 − 0.0121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 0.762 - 0.647i)\)

Particular Values

\(L(15)\) \(\approx\) \(0.07459186121\)
\(L(\frac12)\) \(\approx\) \(0.07459186121\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.39e6 - 4.14e6i)T \)
7 \( 1 + (4.94e11 - 1.72e12i)T \)
good2 \( 1 + (-2.30e4 - 3.98e4i)T + (-2.68e8 + 4.64e8i)T^{2} \)
5 \( 1 + (-1.61e9 - 2.79e9i)T + (-9.31e19 + 1.61e20i)T^{2} \)
11 \( 1 + (-3.88e13 + 6.73e13i)T + (-7.93e29 - 1.37e30i)T^{2} \)
13 \( 1 - 1.49e16T + 2.01e32T^{2} \)
17 \( 1 + (-4.88e15 + 8.46e15i)T + (-2.40e35 - 4.17e35i)T^{2} \)
19 \( 1 + (2.24e18 + 3.88e18i)T + (-6.06e36 + 1.05e37i)T^{2} \)
23 \( 1 + (-2.82e19 - 4.89e19i)T + (-1.54e39 + 2.67e39i)T^{2} \)
29 \( 1 + 1.82e21T + 2.56e42T^{2} \)
31 \( 1 + (-1.10e21 + 1.90e21i)T + (-8.88e42 - 1.53e43i)T^{2} \)
37 \( 1 + (9.35e21 + 1.62e22i)T + (-1.50e45 + 2.60e45i)T^{2} \)
41 \( 1 + 4.62e23T + 5.89e46T^{2} \)
43 \( 1 - 6.65e21T + 2.34e47T^{2} \)
47 \( 1 + (-1.41e24 - 2.44e24i)T + (-1.54e48 + 2.68e48i)T^{2} \)
53 \( 1 + (-6.16e24 + 1.06e25i)T + (-5.04e49 - 8.74e49i)T^{2} \)
59 \( 1 + (1.42e25 - 2.46e25i)T + (-1.13e51 - 1.95e51i)T^{2} \)
61 \( 1 + (7.91e24 + 1.37e25i)T + (-2.97e51 + 5.15e51i)T^{2} \)
67 \( 1 + (-6.66e25 + 1.15e26i)T + (-4.52e52 - 7.82e52i)T^{2} \)
71 \( 1 + 6.15e26T + 4.85e53T^{2} \)
73 \( 1 + (2.11e26 - 3.66e26i)T + (-5.43e53 - 9.41e53i)T^{2} \)
79 \( 1 + (-3.04e27 - 5.26e27i)T + (-5.37e54 + 9.30e54i)T^{2} \)
83 \( 1 + 3.45e27T + 4.50e55T^{2} \)
89 \( 1 + (1.02e28 + 1.77e28i)T + (-1.70e56 + 2.95e56i)T^{2} \)
97 \( 1 + 7.53e28T + 4.13e57T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.74254355295269254520529488829, −11.45409062913091316720046270570, −9.247069256659818196052364481341, −8.416811544621670559726058844560, −6.86871426386736266007669152005, −5.98355938400248909338411003731, −5.13032888386979153503840269018, −3.92684154637293367294694764132, −2.84124791462406806632854570931, −0.01248623254125565945290424696, 1.06907770382108978989214866737, 1.77122182046438045452428958902, 3.25363113788133050411448103025, 4.13865591172302768722473516798, 5.39794808182986299102021507084, 6.55241228342003086017200803461, 8.771887377859655041741894886533, 10.25013212452839934896401926916, 10.92772675708223052433016965555, 12.12750498238856661258306797457

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