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Properties

Label 2-84-84.23-c1-0-8
Degree $2$
Conductor $84$
Sign $0.952 - 0.303i$
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  + (1.09 + 0.893i)2-s + (1.03 − 1.39i)3-s + (0.404 + 1.95i)4-s + (−1.79 − 1.03i)5-s + (2.37 − 0.602i)6-s + (−2.61 + 0.412i)7-s + (−1.30 + 2.50i)8-s + (−0.867 − 2.87i)9-s + (−1.04 − 2.74i)10-s + (1.99 + 3.45i)11-s + (3.14 + 1.46i)12-s − 1.30·13-s + (−3.23 − 1.88i)14-s + (−3.30 + 1.42i)15-s + (−3.67 + 1.58i)16-s + (2.54 − 1.47i)17-s + ⋯
L(s)  = 1  + (0.775 + 0.631i)2-s + (0.596 − 0.802i)3-s + (0.202 + 0.979i)4-s + (−0.804 − 0.464i)5-s + (0.969 − 0.245i)6-s + (−0.987 + 0.155i)7-s + (−0.461 + 0.887i)8-s + (−0.289 − 0.957i)9-s + (−0.330 − 0.867i)10-s + (0.601 + 1.04i)11-s + (0.906 + 0.421i)12-s − 0.363·13-s + (−0.864 − 0.503i)14-s + (−0.852 + 0.368i)15-s + (−0.918 + 0.396i)16-s + (0.617 − 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36021 + 0.211296i\)
\(L(\frac12)\) \(\approx\) \(1.36021 + 0.211296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.893i)T \)
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

−14.42181792960760605366953141646, −13.16851918045286922006906042418, −12.40379134745003789000021227036, −11.86717556217348078906812721510, −9.551510030314156011865947309473, −8.350266477275787045745457331958, −7.28987518388467087431953890908, −6.37680614091547554667261179195, −4.48317053259562261317078561216, −2.99891680439949824354263468846, 3.17854986364983824693333943659, 3.82067024774916718225935356244, 5.59015447249070626029174256865, 7.20880421943641176331933468725, 8.965684262235500872761194848705, 10.02814455571949813992169615707, 11.02121861158940176855928455271, 11.97411442771974375448883667621, 13.36409117035860722514925678066, 14.15011059473626943909902213763

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