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Properties

Label 2-84-12.11-c5-0-27
Degree $2$
Conductor $84$
Sign $0.757 + 0.652i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
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  + (−1.03 − 5.56i)2-s + (13.7 − 7.34i)3-s + (−29.8 + 11.5i)4-s + 59.1i·5-s + (−55.1 − 68.8i)6-s + 49i·7-s + (95.2 + 153. i)8-s + (135. − 201. i)9-s + (329. − 61.4i)10-s + 242.·11-s + (−325. + 377. i)12-s + 886.·13-s + (272. − 50.9i)14-s + (434. + 813. i)15-s + (757. − 689. i)16-s − 988. i·17-s + ⋯
L(s)  = 1  + (−0.183 − 0.982i)2-s + (0.882 − 0.470i)3-s + (−0.932 + 0.361i)4-s + 1.05i·5-s + (−0.624 − 0.780i)6-s + 0.377i·7-s + (0.526 + 0.850i)8-s + (0.556 − 0.830i)9-s + (1.04 − 0.194i)10-s + 0.603·11-s + (−0.652 + 0.757i)12-s + 1.45·13-s + (0.371 − 0.0694i)14-s + (0.498 + 0.933i)15-s + (0.739 − 0.673i)16-s − 0.829i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.04326 - 0.758736i\)
\(L(\frac12)\) \(\approx\) \(2.04326 - 0.758736i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 5.56i)T \)
3 \( 1 + (-13.7 + 7.34i)T \)
7 \( 1 - 49iT \)
good5 \( 1 - 59.1iT - 3.12e3T^{2} \)
11 \( 1 - 242.T + 1.61e5T^{2} \)
13 \( 1 - 886.T + 3.71e5T^{2} \)
17 \( 1 + 988. iT - 1.41e6T^{2} \)
19 \( 1 - 2.55e3iT - 2.47e6T^{2} \)
23 \( 1 - 27.1T + 6.43e6T^{2} \)
29 \( 1 - 924. iT - 2.05e7T^{2} \)
31 \( 1 + 30.5iT - 2.86e7T^{2} \)
37 \( 1 - 1.32e4T + 6.93e7T^{2} \)
41 \( 1 + 9.49e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.13e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.72e4T + 2.29e8T^{2} \)
53 \( 1 - 1.78e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.89e4T + 7.14e8T^{2} \)
61 \( 1 - 3.28e4T + 8.44e8T^{2} \)
67 \( 1 - 3.60e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.83e4T + 1.80e9T^{2} \)
73 \( 1 + 7.38e4T + 2.07e9T^{2} \)
79 \( 1 + 6.55e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.12e4T + 3.93e9T^{2} \)
89 \( 1 + 4.18e4iT - 5.58e9T^{2} \)
97 \( 1 - 3.85e4T + 8.58e9T^{2} \)
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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

−13.17717835805156145834132865113, −12.03937462073189799421523727935, −11.04608776152374624747532746809, −9.852775071929327037449034427960, −8.812691115393577047831962082557, −7.73322744416188930813446350074, −6.24179554323922921464737066071, −3.85397254147007287610688290261, −2.86296070811001385093946477186, −1.41064011626807765453477615904, 1.12379708109284480632934083649, 3.85563289441203898092407224313, 4.86806188023119784974727384152, 6.49041211748792226065644712391, 8.029765170527643402940988055429, 8.784268576746373102201662034261, 9.556877196476095240609404198066, 10.97375251117188018605576354232, 13.02259210184747444295499582885, 13.46355663816515382533884745715

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