Properties

Label 2-21-21.20-c21-0-50
Degree $2$
Conductor $21$
Sign $0.796 - 0.605i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.42e3i·2-s + (−1.00e5 − 1.83e4i)3-s − 3.79e6·4-s + 2.12e7·5-s + (−4.46e7 + 2.44e8i)6-s + (5.03e8 − 5.51e8i)7-s + 4.11e9i·8-s + (9.78e9 + 3.70e9i)9-s − 5.16e10i·10-s − 1.35e11i·11-s + (3.81e11 + 6.97e10i)12-s + 6.90e11i·13-s + (−1.33e12 − 1.22e12i)14-s + (−2.14e12 − 3.91e11i)15-s + 2.02e12·16-s − 7.37e12·17-s + ⋯
L(s)  = 1  − 1.67i·2-s + (−0.983 − 0.179i)3-s − 1.80·4-s + 0.975·5-s + (−0.301 + 1.64i)6-s + (0.674 − 0.738i)7-s + 1.35i·8-s + (0.935 + 0.353i)9-s − 1.63i·10-s − 1.57i·11-s + (1.77 + 0.325i)12-s + 1.39i·13-s + (−1.23 − 1.12i)14-s + (−0.959 − 0.175i)15-s + 0.460·16-s − 0.886·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.796 - 0.605i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ 0.796 - 0.605i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.2614543666\)
\(L(\frac12)\) \(\approx\) \(0.2614543666\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.00e5 + 1.83e4i)T \)
7 \( 1 + (-5.03e8 + 5.51e8i)T \)
good2 \( 1 + 2.42e3iT - 2.09e6T^{2} \)
5 \( 1 - 2.12e7T + 4.76e14T^{2} \)
11 \( 1 + 1.35e11iT - 7.40e21T^{2} \)
13 \( 1 - 6.90e11iT - 2.47e23T^{2} \)
17 \( 1 + 7.37e12T + 6.90e25T^{2} \)
19 \( 1 - 6.66e11iT - 7.14e26T^{2} \)
23 \( 1 + 1.94e13iT - 3.94e28T^{2} \)
29 \( 1 - 8.36e14iT - 5.13e30T^{2} \)
31 \( 1 + 6.31e14iT - 2.08e31T^{2} \)
37 \( 1 + 4.54e16T + 8.55e32T^{2} \)
41 \( 1 + 1.81e16T + 7.38e33T^{2} \)
43 \( 1 + 2.33e17T + 2.00e34T^{2} \)
47 \( 1 - 4.03e17T + 1.30e35T^{2} \)
53 \( 1 + 1.74e18iT - 1.62e36T^{2} \)
59 \( 1 + 2.02e18T + 1.54e37T^{2} \)
61 \( 1 - 3.65e18iT - 3.10e37T^{2} \)
67 \( 1 + 2.52e19T + 2.22e38T^{2} \)
71 \( 1 - 5.48e18iT - 7.52e38T^{2} \)
73 \( 1 - 5.62e19iT - 1.34e39T^{2} \)
79 \( 1 - 8.73e19T + 7.08e39T^{2} \)
83 \( 1 - 2.61e20T + 1.99e40T^{2} \)
89 \( 1 + 2.02e20T + 8.65e40T^{2} \)
97 \( 1 - 9.52e20iT - 5.27e41T^{2} \)
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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

−11.81953063514981834276901545386, −11.07444760549205398436268430620, −10.23244600680746141075362341809, −8.859001026326858024738247764856, −6.60169151881351394434296297006, −5.08136349813339431214574487753, −3.85019328072764356976591360602, −2.02392945933310207554699675896, −1.24816095324680809072768320039, −0.07715278461124913262906409223, 1.85355220043911736544888299676, 4.70864515607101823370679031583, 5.40705937862938509966311753383, 6.38204558028004448648143929716, 7.60137015895034229267356630154, 9.147884559375722096236524044530, 10.37714187854805280494991080945, 12.25778948239339722507934256229, 13.51451911912932419408205091222, 15.13440395946487438274708266475

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