Properties

Label 2-21-21.17-c5-0-4
Degree $2$
Conductor $21$
Sign $0.473 - 0.880i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 + 7.79i)3-s + (−16 + 27.7i)4-s + (105.5 + 75.3i)7-s + (121.5 + 210. i)9-s + (−432 + 249. i)12-s − 1.14e3i·13-s + (−511. − 886. i)16-s + (−139.5 + 80.5i)19-s + (837 + 1.83e3i)21-s + (1.56e3 − 2.70e3i)25-s + 3.78e3i·27-s + (−3.77e3 + 1.71e3i)28-s + (8.96e3 + 5.17e3i)31-s − 7.77e3·36-s + (−3.33e3 − 5.76e3i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.813 + 0.581i)7-s + (0.5 + 0.866i)9-s + (−0.866 + 0.499i)12-s − 1.87i·13-s + (−0.499 − 0.866i)16-s + (−0.0886 + 0.0511i)19-s + (0.414 + 0.910i)21-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (−0.910 + 0.414i)28-s + (1.67 + 0.967i)31-s − 36-s + (−0.399 − 0.692i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.473 - 0.880i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ 0.473 - 0.880i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.46231 + 0.873953i\)
\(L(\frac12)\) \(\approx\) \(1.46231 + 0.873953i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 + (-105.5 - 75.3i)T \)
good2 \( 1 + (16 - 27.7i)T^{2} \)
5 \( 1 + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (139.5 - 80.5i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.05e7T^{2} \)
31 \( 1 + (-8.96e3 - 5.17e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.33e3 + 5.76e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 + 2.24e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (3.76e4 - 2.17e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.89e4 + 3.28e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.80e9T^{2} \)
73 \( 1 + (4.05e4 + 2.33e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.54e4 + 7.86e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.27e5iT - 8.58e9T^{2} \)
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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

−17.45106670097062708324859298614, −15.88889809540984035800331025499, −14.79381128271610411718356820870, −13.51148327349410256042706359072, −12.25056191461997646022441201145, −10.36542466702961823442798597344, −8.675767895599749167662661560049, −7.906434391186503279923112065439, −4.86892729722762588698310935596, −2.99607649272218920312010938558, 1.53673781904941817027995831137, 4.43281804449275052807917454414, 6.78400659045666483863415812808, 8.522745617566955560754995627565, 9.792717799753384105591753804808, 11.52544330452200570644093092456, 13.47662108028634921731768651174, 14.17987293565371911678086144675, 15.17665312391879728702916164679, 17.09380091478748034014376654794

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