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Properties

Label 2-84-28.3-c1-0-6
Degree $2$
Conductor $84$
Sign $0.999 + 0.00458i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.178i)2-s + (0.5 − 0.866i)3-s + (1.93 + 0.502i)4-s + (−3.33 + 1.92i)5-s + (0.856 − 1.12i)6-s + (−1.59 − 2.11i)7-s + (2.62 + 1.05i)8-s + (−0.499 − 0.866i)9-s + (−5.02 + 2.10i)10-s + (−1.17 − 0.681i)11-s + (1.40 − 1.42i)12-s + 0.369i·13-s + (−1.85 − 3.24i)14-s + 3.85i·15-s + (3.49 + 1.94i)16-s + (3.89 + 2.25i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.126i)2-s + (0.288 − 0.499i)3-s + (0.967 + 0.251i)4-s + (−1.49 + 0.862i)5-s + (0.349 − 0.459i)6-s + (−0.602 − 0.798i)7-s + (0.928 + 0.371i)8-s + (−0.166 − 0.288i)9-s + (−1.59 + 0.666i)10-s + (−0.355 − 0.205i)11-s + (0.404 − 0.411i)12-s + 0.102i·13-s + (−0.496 − 0.868i)14-s + 0.995i·15-s + (0.873 + 0.486i)16-s + (0.945 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.999 + 0.00458i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.999 + 0.00458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42762 - 0.00327373i\)
\(L(\frac12)\) \(\approx\) \(1.42762 - 0.00327373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.178i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.59 + 2.11i)T \)
good5 \( 1 + (3.33 - 1.92i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.17 + 0.681i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.369iT - 13T^{2} \)
17 \( 1 + (-3.89 - 2.25i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0330 + 0.0573i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.77 - 1.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.11T + 29T^{2} \)
31 \( 1 + (-3.01 + 5.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.74 - 4.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.45iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 + (-0.712 - 1.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.27 + 2.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.71 - 2.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.23 + 0.715i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.45 + 4.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + (1.56 + 0.900i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.8 - 6.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (-1.11 + 0.646i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.88iT - 97T^{2} \)
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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

−14.28187199026268535889178496708, −13.28300157135359719440478597787, −12.23298718692485555528492237591, −11.34366869153124028578394985838, −10.29521344393314774193512108334, −7.938408514420699269606951245231, −7.37823022536774871140927769589, −6.21075714808866561365474181661, −4.06951855965350634757947622721, −3.13236453059016307079577970554, 3.12681587570989946224126052022, 4.37563289662944674232084119740, 5.54593622847326613476353750602, 7.40573416985436534431848597744, 8.556552438397003773863888102476, 9.998685068533117067445997779806, 11.46781050029700819510826232776, 12.23270148744364212753321456722, 12.97759281849599256952338717296, 14.43748537386821591332499902798

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