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Properties

Label 2-84-4.3-c4-0-9
Degree $2$
Conductor $84$
Sign $0.649 + 0.760i$
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  + (−3.63 − 1.67i)2-s + 5.19i·3-s + (10.3 + 12.1i)4-s − 41.3·5-s + (8.70 − 18.8i)6-s − 18.5i·7-s + (−17.3 − 61.5i)8-s − 27·9-s + (150. + 69.2i)10-s + 41.7i·11-s + (−63.2 + 53.9i)12-s + 260.·13-s + (−31.0 + 67.2i)14-s − 215. i·15-s + (−40.0 + 252. i)16-s + 208.·17-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)2-s + 0.577i·3-s + (0.649 + 0.760i)4-s − 1.65·5-s + (0.241 − 0.524i)6-s − 0.377i·7-s + (−0.271 − 0.962i)8-s − 0.333·9-s + (1.50 + 0.692i)10-s + 0.344i·11-s + (−0.439 + 0.374i)12-s + 1.54·13-s + (−0.158 + 0.343i)14-s − 0.955i·15-s + (−0.156 + 0.987i)16-s + 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.649 + 0.760i$
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: \((2,\ 84,\ (\ :2),\ 0.649 + 0.760i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.650185 - 0.299734i\)
\(L(\frac12)\) \(\approx\) \(0.650185 - 0.299734i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.63 + 1.67i)T \)
3 \( 1 - 5.19iT \)
7 \( 1 + 18.5iT \)
good5 \( 1 + 41.3T + 625T^{2} \)
11 \( 1 - 41.7iT - 1.46e4T^{2} \)
13 \( 1 - 260.T + 2.85e4T^{2} \)
17 \( 1 - 208.T + 8.35e4T^{2} \)
19 \( 1 + 418. iT - 1.30e5T^{2} \)
23 \( 1 + 471. iT - 2.79e5T^{2} \)
29 \( 1 - 835.T + 7.07e5T^{2} \)
31 \( 1 + 760. iT - 9.23e5T^{2} \)
37 \( 1 + 388.T + 1.87e6T^{2} \)
41 \( 1 + 1.40e3T + 2.82e6T^{2} \)
43 \( 1 - 2.10e3iT - 3.41e6T^{2} \)
47 \( 1 + 725. iT - 4.87e6T^{2} \)
53 \( 1 - 2.26e3T + 7.89e6T^{2} \)
59 \( 1 + 6.73e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.63e3T + 1.38e7T^{2} \)
67 \( 1 - 2.70e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.11e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.60e3T + 2.83e7T^{2} \)
79 \( 1 + 1.18e4iT - 3.89e7T^{2} \)
83 \( 1 + 2.65e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.09e4T + 6.27e7T^{2} \)
97 \( 1 + 1.12e4T + 8.85e7T^{2} \)
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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

−13.09335105380406054937060457433, −11.81995275481466707711879578012, −11.17473587985043880732216373077, −10.22722985975045946536407107582, −8.759894092530993091836235150184, −8.010406404086195337226649147300, −6.76810169506126570938593053783, −4.31650483429371171947536827027, −3.27012701082199429423008269370, −0.61473514635534073678139868083, 1.10073740889731933321843712866, 3.48933321793640190966056960640, 5.71354866068371302356866009744, 7.04762600819710439368004095234, 8.140280665897444674623834118288, 8.650773760361380375264342979407, 10.46812411707075405543770500911, 11.55238032730551893842580434193, 12.18758759275189392168718474324, 13.86854739360626196809869717901

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