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Properties

Label 2-84-12.11-c3-0-10
Degree $2$
Conductor $84$
Sign $0.799 - 0.600i$
Analytic cond. $4.95616$
Root an. cond. $2.22624$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 − 2.82i)2-s + (3.69 + 3.65i)3-s + (−7.91 − 1.17i)4-s + 16.0i·5-s + (11.0 − 9.66i)6-s + 7i·7-s + (−4.95 + 22.0i)8-s + (0.298 + 26.9i)9-s + (45.3 + 3.34i)10-s + 10.8·11-s + (−24.9 − 33.2i)12-s − 5.57·13-s + (19.7 + 1.45i)14-s + (−58.7 + 59.4i)15-s + (61.2 + 18.5i)16-s + 13.9i·17-s + ⋯
L(s)  = 1  + (0.0735 − 0.997i)2-s + (0.710 + 0.703i)3-s + (−0.989 − 0.146i)4-s + 1.43i·5-s + (0.753 − 0.657i)6-s + 0.377i·7-s + (−0.218 + 0.975i)8-s + (0.0110 + 0.999i)9-s + (1.43 + 0.105i)10-s + 0.297·11-s + (−0.600 − 0.799i)12-s − 0.118·13-s + (0.376 + 0.0277i)14-s + (−1.01 + 1.02i)15-s + (0.956 + 0.290i)16-s + 0.199i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.799 - 0.600i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.59277 + 0.531154i\)
\(L(\frac12)\) \(\approx\) \(1.59277 + 0.531154i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.207 + 2.82i)T \)
3 \( 1 + (-3.69 - 3.65i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 16.0iT - 125T^{2} \)
11 \( 1 - 10.8T + 1.33e3T^{2} \)
13 \( 1 + 5.57T + 2.19e3T^{2} \)
17 \( 1 - 13.9iT - 4.91e3T^{2} \)
19 \( 1 + 143. iT - 6.85e3T^{2} \)
23 \( 1 - 192.T + 1.21e4T^{2} \)
29 \( 1 - 210. iT - 2.43e4T^{2} \)
31 \( 1 + 143. iT - 2.97e4T^{2} \)
37 \( 1 + 221.T + 5.06e4T^{2} \)
41 \( 1 + 264. iT - 6.89e4T^{2} \)
43 \( 1 - 355. iT - 7.95e4T^{2} \)
47 \( 1 - 97.7T + 1.03e5T^{2} \)
53 \( 1 + 95.4iT - 1.48e5T^{2} \)
59 \( 1 - 142.T + 2.05e5T^{2} \)
61 \( 1 - 734.T + 2.26e5T^{2} \)
67 \( 1 + 626. iT - 3.00e5T^{2} \)
71 \( 1 - 774.T + 3.57e5T^{2} \)
73 \( 1 - 777.T + 3.89e5T^{2} \)
79 \( 1 + 260. iT - 4.93e5T^{2} \)
83 \( 1 + 768.T + 5.71e5T^{2} \)
89 \( 1 + 397. iT - 7.04e5T^{2} \)
97 \( 1 + 774.T + 9.12e5T^{2} \)
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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

−13.97335408003270014411100503772, −12.89943190822375440095039427850, −11.24163040483435553379874277465, −10.80090129337395378366711895308, −9.597201841649759128673774302998, −8.711689219976943188960096130407, −7.03764909136609403715774721940, −5.02389460881246548070827611573, −3.42714804303345768687236083789, −2.50231039128066916044640257674, 1.04339496075592723962814054498, 3.89364438779747940277373053923, 5.35111380825039731621687652619, 6.83514278485079803724869194533, 8.033943892755434022913258219281, 8.775248309413362347388296636597, 9.776845012883768005933593194207, 12.12133182400892842171134869352, 12.86787292129400650646885610892, 13.70507856704540649334638750876

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