Properties

Label 2-84-28.27-c11-0-82
Degree $2$
Conductor $84$
Sign $-0.0397 - 0.999i$
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  + (16.3 − 42.1i)2-s − 243·3-s + (−1.51e3 − 1.37e3i)4-s − 1.05e4i·5-s + (−3.97e3 + 1.02e4i)6-s + (3.12e4 + 3.16e4i)7-s + (−8.29e4 + 4.13e4i)8-s + 5.90e4·9-s + (−4.44e5 − 1.72e5i)10-s − 9.07e5i·11-s + (3.67e5 + 3.35e5i)12-s − 7.60e5i·13-s + (1.84e6 − 8.00e5i)14-s + 2.55e6i·15-s + (3.87e5 + 4.17e6i)16-s − 4.72e6i·17-s + ⋯
L(s)  = 1  + (0.361 − 0.932i)2-s − 0.577·3-s + (−0.739 − 0.673i)4-s − 1.50i·5-s + (−0.208 + 0.538i)6-s + (0.702 + 0.711i)7-s + (−0.895 + 0.445i)8-s + 0.333·9-s + (−1.40 − 0.544i)10-s − 1.69i·11-s + (0.426 + 0.388i)12-s − 0.568i·13-s + (0.917 − 0.397i)14-s + 0.869i·15-s + (0.0922 + 0.995i)16-s − 0.807i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0397 - 0.999i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0397 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.0397 - 0.999i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.0397 - 0.999i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.854885 + 0.889538i\)
\(L(\frac12)\) \(\approx\) \(0.854885 + 0.889538i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16.3 + 42.1i)T \)
3 \( 1 + 243T \)
7 \( 1 + (-3.12e4 - 3.16e4i)T \)
good5 \( 1 + 1.05e4iT - 4.88e7T^{2} \)
11 \( 1 + 9.07e5iT - 2.85e11T^{2} \)
13 \( 1 + 7.60e5iT - 1.79e12T^{2} \)
17 \( 1 + 4.72e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.21e7T + 1.16e14T^{2} \)
23 \( 1 + 5.18e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.09e8T + 1.22e16T^{2} \)
31 \( 1 - 1.85e8T + 2.54e16T^{2} \)
37 \( 1 + 1.55e8T + 1.77e17T^{2} \)
41 \( 1 - 1.09e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.46e9iT - 9.29e17T^{2} \)
47 \( 1 - 2.04e9T + 2.47e18T^{2} \)
53 \( 1 + 9.46e8T + 9.26e18T^{2} \)
59 \( 1 - 3.46e9T + 3.01e19T^{2} \)
61 \( 1 - 1.24e10iT - 4.35e19T^{2} \)
67 \( 1 - 9.80e9iT - 1.22e20T^{2} \)
71 \( 1 + 5.49e9iT - 2.31e20T^{2} \)
73 \( 1 - 1.15e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.91e10iT - 7.47e20T^{2} \)
83 \( 1 + 4.80e9T + 1.28e21T^{2} \)
89 \( 1 + 2.32e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.91e10iT - 7.15e21T^{2} \)
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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

−11.42879316541931833542395485472, −10.43540437390026973312450140389, −8.823332119709560107434495120305, −8.487157435896443455395527975266, −5.91435123240980403791515124293, −5.20612227898783834535883888540, −4.21214041039997967335272276459, −2.46974561377520219676550167321, −1.00282590963199597815528118085, −0.34393307552705214026072699871, 1.90320984815556719831965612457, 3.76339759456424713092619013018, 4.71708955599425499567150981133, 6.23893392180893884450703425324, 7.07653702015659252429804858775, 7.77688091662773501929637824791, 9.659033545870826579296343141010, 10.68529611368389119336694565358, 11.74961384330411617397743798087, 12.99395061247470594181703879395

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