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Properties

Label 2-84-12.11-c1-0-7
Degree $2$
Conductor $84$
Sign $0.998 + 0.0608i$
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  + (1.34 + 0.430i)2-s + (−0.916 − 1.46i)3-s + (1.62 + 1.15i)4-s + 0.348i·5-s + (−0.602 − 2.37i)6-s − i·7-s + (1.69 + 2.26i)8-s + (−1.31 + 2.69i)9-s + (−0.150 + 0.469i)10-s − 3.90·11-s + (0.210 − 3.45i)12-s − 2.93·13-s + (0.430 − 1.34i)14-s + (0.512 − 0.319i)15-s + (1.30 + 3.77i)16-s − 3.90i·17-s + ⋯
L(s)  = 1  + (0.952 + 0.304i)2-s + (−0.529 − 0.848i)3-s + (0.814 + 0.579i)4-s + 0.155i·5-s + (−0.245 − 0.969i)6-s − 0.377i·7-s + (0.599 + 0.800i)8-s + (−0.439 + 0.898i)9-s + (−0.0474 + 0.148i)10-s − 1.17·11-s + (0.0608 − 0.998i)12-s − 0.815·13-s + (0.115 − 0.360i)14-s + (0.132 − 0.0825i)15-s + (0.327 + 0.944i)16-s − 0.946i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.998 + 0.0608i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.998 + 0.0608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31965 - 0.0402016i\)
\(L(\frac12)\) \(\approx\) \(1.31965 - 0.0402016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.430i)T \)
3 \( 1 + (0.916 + 1.46i)T \)
7 \( 1 + iT \)
good5 \( 1 - 0.348iT - 5T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 - 5.57iT - 19T^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 + 9.75iT - 29T^{2} \)
31 \( 1 + 2.63iT - 31T^{2} \)
37 \( 1 - 0.639T + 37T^{2} \)
41 \( 1 - 7.57iT - 41T^{2} \)
43 \( 1 + 2.51iT - 43T^{2} \)
47 \( 1 - 4.36T + 47T^{2} \)
53 \( 1 - 1.72iT - 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 + 0.639iT - 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 7.87T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−14.02981973548596367676632714643, −13.22235648863063013993121403244, −12.33942199242103248595506112614, −11.38457465708374488297603935742, −10.25994119439235670400913770433, −7.956038828687021013273706169893, −7.24717638784428222116721073743, −5.94396641703700883856860150464, −4.78771429570704647593834017756, −2.62369144497836808755406868877, 2.96303196562227100648239238537, 4.71684099634878143413359447914, 5.47885126754850771673006861106, 6.98465057703508213337928389440, 8.955071294499425078885367380622, 10.36700447579834258745451868445, 10.98410854322699648999967949636, 12.27672108807410175720734681211, 12.98037529658177347876485456420, 14.47228109343457678568331218541

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