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Properties

Label 2-21-7.5-c2-0-2
Degree $2$
Conductor $21$
Sign $0.795 + 0.605i$
Analytic cond. $0.572208$
Root an. cond. $0.756444$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.5 + 0.866i)3-s + (−3 − 1.73i)5-s + 3.46i·6-s + (−3.5 + 6.06i)7-s + 8·8-s + (1.5 − 2.59i)9-s + (−6 + 3.46i)10-s + (−5 − 8.66i)11-s − 12.1i·13-s + (7 + 12.1i)14-s + 6·15-s + (8 − 13.8i)16-s + (−6 + 3.46i)17-s + (−3 − 5.19i)18-s + (28.5 + 16.4i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.288i)3-s + (−0.600 − 0.346i)5-s + 0.577i·6-s + (−0.5 + 0.866i)7-s + 8-s + (0.166 − 0.288i)9-s + (−0.600 + 0.346i)10-s + (−0.454 − 0.787i)11-s − 0.932i·13-s + (0.5 + 0.866i)14-s + 0.400·15-s + (0.5 − 0.866i)16-s + (−0.352 + 0.203i)17-s + (−0.166 − 0.288i)18-s + (1.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(0.572208\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1),\ 0.795 + 0.605i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.890919 - 0.300366i\)
\(L(\frac12)\) \(\approx\) \(0.890919 - 0.300366i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (3.5 - 6.06i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (3 + 1.73i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5 + 8.66i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 12.1iT - 169T^{2} \)
17 \( 1 + (6 - 3.46i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-28.5 - 16.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (20 - 34.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 16T + 841T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 24.2iT - 1.68e3T^{2} \)
43 \( 1 + 19T + 1.84e3T^{2} \)
47 \( 1 + (45 + 25.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-16 - 27.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-36 + 20.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18 - 10.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.5 + 51.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 26T + 5.04e3T^{2} \)
73 \( 1 + (16.5 - 9.52i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (23.5 - 40.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 24.2iT - 6.88e3T^{2} \)
89 \( 1 + (-102 - 58.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 48.4iT - 9.40e3T^{2} \)
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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

−17.98575500164893497617612111575, −16.28447276859396086641175184325, −15.63988802368905518660367202192, −13.54432850756280588224298023507, −12.27880611898751037709276697879, −11.55461121444636327817579051007, −10.06348780879576766014782917988, −8.017509227007492090205306035738, −5.50551512778034746268598153449, −3.43300208294605646978948795414, 4.61368157550132321146797149827, 6.62750552293455248086161610210, 7.47388105315423938103714340800, 10.13928041292070285369187287890, 11.53885667903414972455922494576, 13.22209945964985216508732192582, 14.34528053835078955911414753800, 15.75797551673013038124459084352, 16.46597840674563552948280763401, 17.90683416423905283373161651990

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