Properties

Label 2-84-84.23-c1-0-7
Degree $2$
Conductor $84$
Sign $-0.358 + 0.933i$
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  + (−0.951 + 1.04i)2-s + (−1.66 − 0.475i)3-s + (−0.190 − 1.99i)4-s + (−2.36 − 1.36i)5-s + (2.08 − 1.29i)6-s + (−1.89 − 1.84i)7-s + (2.26 + 1.69i)8-s + (2.54 + 1.58i)9-s + (3.67 − 1.17i)10-s + (−1.02 − 1.76i)11-s + (−0.629 + 3.40i)12-s − 4.44·13-s + (3.73 − 0.232i)14-s + (3.28 + 3.39i)15-s + (−3.92 + 0.759i)16-s + (0.571 − 0.330i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.740i)2-s + (−0.961 − 0.274i)3-s + (−0.0953 − 0.995i)4-s + (−1.05 − 0.609i)5-s + (0.850 − 0.526i)6-s + (−0.717 − 0.696i)7-s + (0.800 + 0.598i)8-s + (0.849 + 0.528i)9-s + (1.16 − 0.371i)10-s + (−0.307 − 0.532i)11-s + (−0.181 + 0.983i)12-s − 1.23·13-s + (0.998 − 0.0621i)14-s + (0.848 + 0.876i)15-s + (−0.981 + 0.189i)16-s + (0.138 − 0.0800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.358 + 0.933i$
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: \((2,\ 84,\ (\ :1/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129302 - 0.188199i\)
\(L(\frac12)\) \(\approx\) \(0.129302 - 0.188199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 1.04i)T \)
3 \( 1 + (1.66 + 0.475i)T \)
7 \( 1 + (1.89 + 1.84i)T \)
good5 \( 1 + (2.36 + 1.36i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.02 + 1.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + (-0.571 + 0.330i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.54 - 1.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.521 - 0.902i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.06iT - 29T^{2} \)
31 \( 1 + (-2.68 + 1.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.76 + 8.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 - 0.530iT - 43T^{2} \)
47 \( 1 + (-0.521 + 0.902i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.16 - 5.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.74 + 3.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0992 + 0.171i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.12 - 2.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 + (-1.51 - 2.62i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.45 + 5.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.15T + 83T^{2} \)
89 \( 1 + (-0.468 - 0.270i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−14.01797074911488834338706685719, −12.76711563904358168024687187285, −11.72677075544670355869658786186, −10.57137686183606866139053829192, −9.527133299110465595441481523914, −7.87323897659061769395350170769, −7.21630074724585026742839342327, −5.79027101294043872916292763649, −4.44836422611681104490021284289, −0.38989777422332225384693281936, 3.06372709978767137127841090385, 4.70673043403025851114978044802, 6.73330909686000633933326663238, 7.77521973762259366965971498638, 9.486901898008324671721528379356, 10.26619652462212347995236309093, 11.48600519683683376007117569724, 12.06379839119806584023692328078, 12.92391662817802070954498010258, 14.98823111221846710588514583196

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