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Properties

Label 2-84-84.11-c1-0-10
Degree $2$
Conductor $84$
Sign $0.583 + 0.812i$
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  + (1.32 − 0.502i)2-s + (−1.03 − 1.39i)3-s + (1.49 − 1.32i)4-s + (−1.79 + 1.03i)5-s + (−2.06 − 1.31i)6-s + (2.61 + 0.412i)7-s + (1.30 − 2.50i)8-s + (−0.867 + 2.87i)9-s + (−1.85 + 2.27i)10-s + (−1.99 + 3.45i)11-s + (−3.39 − 0.704i)12-s − 1.30·13-s + (3.66 − 0.769i)14-s + (3.30 + 1.42i)15-s + (0.464 − 3.97i)16-s + (2.54 + 1.47i)17-s + ⋯
L(s)  = 1  + (0.934 − 0.355i)2-s + (−0.596 − 0.802i)3-s + (0.747 − 0.664i)4-s + (−0.804 + 0.464i)5-s + (−0.842 − 0.538i)6-s + (0.987 + 0.155i)7-s + (0.461 − 0.887i)8-s + (−0.289 + 0.957i)9-s + (−0.586 + 0.719i)10-s + (−0.601 + 1.04i)11-s + (−0.979 − 0.203i)12-s − 0.363·13-s + (0.978 − 0.205i)14-s + (0.852 + 0.368i)15-s + (0.116 − 0.993i)16-s + (0.617 + 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.583 + 0.812i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.583 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12393 - 0.576715i\)
\(L(\frac12)\) \(\approx\) \(1.12393 - 0.576715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.502i)T \)
3 \( 1 + (1.03 + 1.39i)T \)
7 \( 1 + (-2.61 - 0.412i)T \)
good5 \( 1 + (1.79 - 1.03i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.99 - 3.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 + (-2.54 - 1.47i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.949 - 0.548i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.75 + 6.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.865iT - 29T^{2} \)
31 \( 1 + (3.18 + 1.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.08 + 3.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.01iT - 41T^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 + (-3.75 - 6.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.27 - 2.47i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 4.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.05 + 3.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.09 - 1.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.901T + 71T^{2} \)
73 \( 1 + (6.50 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.97 - 5.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (7.38 - 4.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−14.16569797575414661916342446723, −12.70128677170244093319184847741, −12.15269711504205653079094385541, −11.18092517166065853338994216539, −10.34334875030359555471982391033, −7.913726397005701343322102154506, −7.11903469491635677287706725536, −5.62103242995508287258070500364, −4.37708335600163990185225146390, −2.19778816236433726332361344958, 3.59072180711244618094878591963, 4.81445450531577367086887640872, 5.70802727983766407668996696329, 7.52726402729842608520649590409, 8.557252472157004240417196041100, 10.46262186330826012670318708868, 11.55759693277248839070368262062, 12.00424615336386158141137860124, 13.51042562767080563330085553075, 14.59131562636190816146780705991

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