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Properties

Label 2-84-28.19-c3-0-16
Degree $2$
Conductor $84$
Sign $0.801 + 0.597i$
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  + (−0.0644 + 2.82i)2-s + (−1.5 − 2.59i)3-s + (−7.99 − 0.364i)4-s + (−4.32 − 2.49i)5-s + (7.44 − 4.07i)6-s + (15.5 − 10.1i)7-s + (1.54 − 22.5i)8-s + (−4.5 + 7.79i)9-s + (7.34 − 12.0i)10-s + (39.8 − 23.0i)11-s + (11.0 + 21.3i)12-s − 88.6i·13-s + (27.6 + 44.4i)14-s + 14.9i·15-s + (63.7 + 5.82i)16-s + (−86.9 + 50.2i)17-s + ⋯
L(s)  = 1  + (−0.0227 + 0.999i)2-s + (−0.288 − 0.499i)3-s + (−0.998 − 0.0455i)4-s + (−0.387 − 0.223i)5-s + (0.506 − 0.277i)6-s + (0.837 − 0.547i)7-s + (0.0682 − 0.997i)8-s + (−0.166 + 0.288i)9-s + (0.232 − 0.381i)10-s + (1.09 − 0.630i)11-s + (0.265 + 0.512i)12-s − 1.89i·13-s + (0.527 + 0.849i)14-s + 0.258i·15-s + (0.995 + 0.0909i)16-s + (−1.24 + 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.801 + 0.597i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.801 + 0.597i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03830 - 0.344490i\)
\(L(\frac12)\) \(\approx\) \(1.03830 - 0.344490i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0644 - 2.82i)T \)
3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (-15.5 + 10.1i)T \)
good5 \( 1 + (4.32 + 2.49i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-39.8 + 23.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 88.6iT - 2.19e3T^{2} \)
17 \( 1 + (86.9 - 50.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-4.47 + 7.75i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-47.8 - 27.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 96.5T + 2.43e4T^{2} \)
31 \( 1 + (125. + 216. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (159. - 276. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 65.9iT - 6.89e4T^{2} \)
43 \( 1 + 55.9iT - 7.95e4T^{2} \)
47 \( 1 + (-146. + 253. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-74.3 - 128. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-119. - 206. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-597. - 345. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-575. + 332. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.02e3iT - 3.57e5T^{2} \)
73 \( 1 + (-251. + 145. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-173. - 100. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 409.T + 5.71e5T^{2} \)
89 \( 1 + (-91.6 - 52.9i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 301. iT - 9.12e5T^{2} \)
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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

−13.70893680388599281184943069742, −12.90496171802850656914060858423, −11.57495592660144492281948142302, −10.37630824860015266104019192760, −8.625080248123726576904053419786, −7.943611650104223358981928076660, −6.70547490595157215550773194466, −5.45647260129955161325193387711, −4.02690414081283509855569658911, −0.75840101505453992211111743762, 1.92220163136786090469645100770, 3.99122629532779565027976049628, 4.92718292136699267669818071871, 6.91421307914145690315073759486, 8.843963077009536935432684818266, 9.373688278973023697848210151558, 11.00659903403709215839227077481, 11.54973476362529443946993830548, 12.34252725443653622770873808165, 14.03618023729663770501319070448

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