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Properties

Label 2-84-28.3-c1-0-2
Degree $2$
Conductor $84$
Sign $0.204 - 0.978i$
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  + (0.850 + 1.12i)2-s + (−0.5 + 0.866i)3-s + (−0.553 + 1.92i)4-s + (0.834 − 0.481i)5-s + (−1.40 + 0.171i)6-s + (−1.20 − 2.35i)7-s + (−2.64 + 1.00i)8-s + (−0.499 − 0.866i)9-s + (1.25 + 0.533i)10-s + (4.74 + 2.74i)11-s + (−1.38 − 1.44i)12-s − 3.75i·13-s + (1.64 − 3.36i)14-s + 0.963i·15-s + (−3.38 − 2.12i)16-s + (−0.594 − 0.343i)17-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)2-s + (−0.288 + 0.499i)3-s + (−0.276 + 0.960i)4-s + (0.373 − 0.215i)5-s + (−0.573 + 0.0700i)6-s + (−0.453 − 0.891i)7-s + (−0.934 + 0.356i)8-s + (−0.166 − 0.288i)9-s + (0.396 + 0.168i)10-s + (1.43 + 0.826i)11-s + (−0.400 − 0.415i)12-s − 1.04i·13-s + (0.438 − 0.898i)14-s + 0.248i·15-s + (−0.846 − 0.531i)16-s + (−0.144 − 0.0832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.204 - 0.978i$
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: \((2,\ 84,\ (\ :1/2),\ 0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901167 + 0.732369i\)
\(L(\frac12)\) \(\approx\) \(0.901167 + 0.732369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 - 1.12i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.20 + 2.35i)T \)
good5 \( 1 + (-0.834 + 0.481i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.74 - 2.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + (0.594 + 0.343i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.44 + 4.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.07 - 0.620i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + (2.41 - 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 - 2.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + (1.80 + 3.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.04 + 3.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.34 - 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.01 + 5.20i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.17 - 4.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (5.76 + 3.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 0.707i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.543T + 83T^{2} \)
89 \( 1 + (-0.480 + 0.277i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−14.66000853670635082592881585361, −13.49893133365043778622683676855, −12.68222743155963042098115350628, −11.42636560185512385382373144098, −9.964443132429470133609176157022, −8.947520537440876629352111470608, −7.30442340827483876539376437492, −6.29472772579348900009523934977, −4.86017595218788734689164314285, −3.65590061496151109908035932221, 2.04414996328470497698590750981, 3.90594977194608269493259062984, 5.86170449088354877670974973912, 6.47454733676483741383720411323, 8.766054310687074959106044514049, 9.719357686286077052991022998612, 11.18368446178948136181035057518, 11.93434245158979375059873967077, 12.80211851680388109697146897544, 14.01972058670626040580803327868

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