Properties

Label 2-84-21.5-c11-0-22
Degree $2$
Conductor $84$
Sign $0.383 + 0.923i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (364.5 − 210. i)3-s + (3.84e4 + 2.23e4i)7-s + (8.85e4 − 1.53e5i)9-s − 1.45e6i·13-s + (−5.63e6 − 3.25e6i)19-s + (1.87e7 + 5.63e4i)21-s + (2.44e7 + 4.22e7i)25-s − 7.45e7i·27-s + (2.73e8 − 1.57e8i)31-s + (5.96e7 − 1.03e8i)37-s + (−3.05e8 − 5.29e8i)39-s + 2.18e8·43-s + (9.78e8 + 1.71e9i)49-s − 2.73e9·57-s + (−1.07e10 − 6.21e9i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.864 + 0.502i)7-s + (0.5 − 0.866i)9-s − 1.08i·13-s + (−0.522 − 0.301i)19-s + (0.999 + 0.00301i)21-s + (0.499 + 0.866i)25-s − 0.999i·27-s + (1.71 − 0.989i)31-s + (0.141 − 0.245i)37-s + (−0.542 − 0.940i)39-s + 0.227·43-s + (0.494 + 0.869i)49-s − 0.602·57-s + (−1.63 − 0.941i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.383 + 0.923i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ 0.383 + 0.923i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.74648 - 1.83288i\)
\(L(\frac12)\) \(\approx\) \(2.74648 - 1.83288i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-364.5 + 210. i)T \)
7 \( 1 + (-3.84e4 - 2.23e4i)T \)
good5 \( 1 + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 1.45e6iT - 1.79e12T^{2} \)
17 \( 1 + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (5.63e6 + 3.25e6i)T + (5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 1.22e16T^{2} \)
31 \( 1 + (-2.73e8 + 1.57e8i)T + (1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-5.96e7 + 1.03e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 5.50e17T^{2} \)
43 \( 1 - 2.18e8T + 9.29e17T^{2} \)
47 \( 1 + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (1.07e10 + 6.21e9i)T + (2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (2.97e9 + 5.15e9i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 - 2.31e20T^{2} \)
73 \( 1 + (-2.75e10 + 1.59e10i)T + (1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (-2.71e10 + 4.70e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + 1.28e21T^{2} \)
89 \( 1 + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 + 1.26e11iT - 7.15e21T^{2} \)
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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

−11.99530083699249016661076669644, −10.75048717822967744100274847660, −9.383785206821784841484280762544, −8.339808346975697969400086992909, −7.59348123397043305801973336736, −6.14140949834141801708416814708, −4.69554157909811064437304811277, −3.14195852561976817387749498775, −2.04907110796179886054655932273, −0.77120180215953608406599383453, 1.29760995943721778137097901463, 2.51878079443057464172103616509, 4.03725895555052673700095112863, 4.82905951358509689272044913458, 6.70654117064118631774547790776, 7.980700800854387752275336793228, 8.802277371994578979957015976349, 10.03842993332988248886229455506, 10.94959930685214184721803372894, 12.19939531135803608820017532140

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