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Properties

Label 2-84-12.11-c3-0-28
Degree $2$
Conductor $84$
Sign $0.439 + 0.898i$
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  + (2.16 − 1.81i)2-s + (3.06 + 4.19i)3-s + (1.40 − 7.87i)4-s − 20.4i·5-s + (14.2 + 3.52i)6-s + 7i·7-s + (−11.2 − 19.6i)8-s + (−8.20 + 25.7i)9-s + (−37.1 − 44.3i)10-s + 32.3·11-s + (37.3 − 18.2i)12-s + 24.7·13-s + (12.7 + 15.1i)14-s + (85.8 − 62.7i)15-s + (−60.0 − 22.0i)16-s + 103. i·17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.590 + 0.807i)3-s + (0.175 − 0.984i)4-s − 1.83i·5-s + (0.970 + 0.240i)6-s + 0.377i·7-s + (−0.497 − 0.867i)8-s + (−0.303 + 0.952i)9-s + (−1.17 − 1.40i)10-s + 0.887·11-s + (0.898 − 0.439i)12-s + 0.527·13-s + (0.242 + 0.289i)14-s + (1.47 − 1.08i)15-s + (−0.938 − 0.344i)16-s + 1.48i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.439 + 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.23883 - 1.39713i\)
\(L(\frac12)\) \(\approx\) \(2.23883 - 1.39713i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.16 + 1.81i)T \)
3 \( 1 + (-3.06 - 4.19i)T \)
7 \( 1 - 7iT \)
good5 \( 1 + 20.4iT - 125T^{2} \)
11 \( 1 - 32.3T + 1.33e3T^{2} \)
13 \( 1 - 24.7T + 2.19e3T^{2} \)
17 \( 1 - 103. iT - 4.91e3T^{2} \)
19 \( 1 - 3.49iT - 6.85e3T^{2} \)
23 \( 1 - 18.3T + 1.21e4T^{2} \)
29 \( 1 + 146. iT - 2.43e4T^{2} \)
31 \( 1 - 326. iT - 2.97e4T^{2} \)
37 \( 1 - 37.6T + 5.06e4T^{2} \)
41 \( 1 - 111. iT - 6.89e4T^{2} \)
43 \( 1 + 119. iT - 7.95e4T^{2} \)
47 \( 1 - 145.T + 1.03e5T^{2} \)
53 \( 1 + 386. iT - 1.48e5T^{2} \)
59 \( 1 - 108.T + 2.05e5T^{2} \)
61 \( 1 + 305.T + 2.26e5T^{2} \)
67 \( 1 - 508. iT - 3.00e5T^{2} \)
71 \( 1 - 99.6T + 3.57e5T^{2} \)
73 \( 1 + 478.T + 3.89e5T^{2} \)
79 \( 1 + 887. iT - 4.93e5T^{2} \)
83 \( 1 - 26.5T + 5.71e5T^{2} \)
89 \( 1 + 807. iT - 7.04e5T^{2} \)
97 \( 1 - 1.29e3T + 9.12e5T^{2} \)
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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

−13.44480887679980154193289979563, −12.61920265279812531379522530719, −11.64108456147785031488594031772, −10.25630120297532286640451612346, −9.115899314668314946511547250674, −8.474346461177702836747746398756, −5.88476341596555632707010313845, −4.72026862935323924448425834079, −3.75717471232301240538865223635, −1.57716272302113645878034162113, 2.65553245235578808629773357056, 3.75635938897920182613074682828, 6.15326008540191995180315581846, 6.97580189231654458153925882662, 7.67615938197768359822455421518, 9.301540638230577911005136297101, 11.09021022851261240745937372107, 11.90174347636782443341628088639, 13.39452366176522346680257983742, 14.07524958977226603991603562014

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