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Properties

Label 2-84-28.3-c1-0-0
Degree $2$
Conductor $84$
Sign $-0.855 - 0.517i$
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  + (−0.546 + 1.30i)2-s + (−0.5 + 0.866i)3-s + (−1.40 − 1.42i)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 + (−0.690 − 5.40i)10-s + (1.17 + 0.681i)11-s + (1.93 − 0.502i)12-s + 0.369i·13-s + (−3.62 + 0.923i)14-s − 3.85i·15-s + (−0.0640 + 3.99i)16-s + (3.89 + 2.25i)17-s + ⋯
L(s)  = 1  + (−0.386 + 0.922i)2-s + (−0.288 + 0.499i)3-s + (−0.701 − 0.712i)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 + (−0.218 − 1.71i)10-s + (0.355 + 0.205i)11-s + (0.558 − 0.144i)12-s + 0.102i·13-s + (−0.969 + 0.246i)14-s − 0.995i·15-s + (−0.0160 + 0.999i)16-s + (0.945 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.855 - 0.517i$
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: \((2,\ 84,\ (\ :1/2),\ -0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151847 + 0.544214i\)
\(L(\frac12)\) \(\approx\) \(0.151847 + 0.544214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.546 - 1.30i)T \)
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

−14.93274108519178507955075462586, −14.37971141887090676452656571127, −12.35919868597485380749940600314, −11.33429759334208263186728079013, −10.36954200874338805258411346892, −8.881855270378190306077556638528, −7.88687843598501902043543816548, −6.78208210969383974135464162230, −5.26861307111125023510256063710, −3.82782947239650031029925458891, 0.938493665518165874507254169903, 3.70539036095209240374571888226, 4.90177724160885104452297277009, 7.51224669935460926617653762445, 8.026593980869380322179753172266, 9.365270665827275976532390798742, 11.02834662333599580496100942484, 11.58136610346417553228058265542, 12.49424903669101243615606874819, 13.42134220261031249466325130068

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