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Properties

Label 2-84-4.3-c4-0-22
Degree $2$
Conductor $84$
Sign $-0.802 + 0.596i$
Analytic cond. $8.68307$
Root an. cond. $2.94670$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.79 − 1.25i)2-s − 5.19i·3-s + (12.8 − 9.53i)4-s − 40.3·5-s + (−6.52 − 19.7i)6-s − 18.5i·7-s + (36.8 − 52.3i)8-s − 27·9-s + (−153. + 50.6i)10-s − 150. i·11-s + (−49.5 − 66.7i)12-s − 230.·13-s + (−23.2 − 70.3i)14-s + 209. i·15-s + (74.0 − 245. i)16-s + 465.·17-s + ⋯
L(s)  = 1  + (0.949 − 0.313i)2-s − 0.577i·3-s + (0.802 − 0.596i)4-s − 1.61·5-s + (−0.181 − 0.548i)6-s − 0.377i·7-s + (0.575 − 0.818i)8-s − 0.333·9-s + (−1.53 + 0.506i)10-s − 1.24i·11-s + (−0.344 − 0.463i)12-s − 1.36·13-s + (−0.118 − 0.358i)14-s + 0.931i·15-s + (0.289 − 0.957i)16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.802 + 0.596i$
Analytic conductor: \(8.68307\)
Root analytic conductor: \(2.94670\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :2),\ -0.802 + 0.596i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.577919 - 1.74784i\)
\(L(\frac12)\) \(\approx\) \(0.577919 - 1.74784i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.79 + 1.25i)T \)
3 \( 1 + 5.19iT \)
7 \( 1 + 18.5iT \)
good5 \( 1 + 40.3T + 625T^{2} \)
11 \( 1 + 150. iT - 1.46e4T^{2} \)
13 \( 1 + 230.T + 2.85e4T^{2} \)
17 \( 1 - 465.T + 8.35e4T^{2} \)
19 \( 1 - 473. iT - 1.30e5T^{2} \)
23 \( 1 + 629. iT - 2.79e5T^{2} \)
29 \( 1 + 51.5T + 7.07e5T^{2} \)
31 \( 1 + 746. iT - 9.23e5T^{2} \)
37 \( 1 - 1.20e3T + 1.87e6T^{2} \)
41 \( 1 - 2.36e3T + 2.82e6T^{2} \)
43 \( 1 - 1.07e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.32e3iT - 4.87e6T^{2} \)
53 \( 1 + 705.T + 7.89e6T^{2} \)
59 \( 1 - 366. iT - 1.21e7T^{2} \)
61 \( 1 - 740.T + 1.38e7T^{2} \)
67 \( 1 - 6.14e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.90e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.06e3T + 2.83e7T^{2} \)
79 \( 1 + 6.13e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.21e4iT - 4.74e7T^{2} \)
89 \( 1 + 7.81e3T + 6.27e7T^{2} \)
97 \( 1 + 916.T + 8.85e7T^{2} \)
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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

−12.85595914621016434033953805186, −12.09234279041214866891847271959, −11.40587850979814262413877285541, −10.19042324832952701244789088731, −8.055118783970897220301340177399, −7.35207304152615935936611550019, −5.81962014261708063710706340796, −4.21682605607157257186672277109, −3.04065219977471219757055990617, −0.67443699031854759414846694596, 2.95067421306705721601883856343, 4.29346996603247325353705624635, 5.19623909082886273710527969515, 7.20343974911163612699739346254, 7.83431152166338658736518177010, 9.564693824600222092274652287747, 11.10178875323802794268103754815, 12.06294378461710485326917297176, 12.56021423633834091598507195926, 14.34974644614976910537853039704

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