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Properties

Label 2-84-28.27-c1-0-6
Degree $2$
Conductor $84$
Sign $0.608 + 0.793i$
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  + (0.780 − 1.17i)2-s + 3-s + (−0.780 − 1.84i)4-s + 1.69i·5-s + (0.780 − 1.17i)6-s + (−2.56 + 0.662i)7-s + (−2.78 − 0.516i)8-s + 9-s + (2 + 1.32i)10-s + 3.02i·11-s + (−0.780 − 1.84i)12-s − 6.04i·13-s + (−1.21 + 3.53i)14-s + 1.69i·15-s + (−2.78 + 2.87i)16-s + 4.34i·17-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + 0.577·3-s + (−0.390 − 0.920i)4-s + 0.758i·5-s + (0.318 − 0.481i)6-s + (−0.968 + 0.250i)7-s + (−0.983 − 0.182i)8-s + 0.333·9-s + (0.632 + 0.418i)10-s + 0.910i·11-s + (−0.225 − 0.531i)12-s − 1.67i·13-s + (−0.325 + 0.945i)14-s + 0.437i·15-s + (−0.695 + 0.718i)16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15050 - 0.567711i\)
\(L(\frac12)\) \(\approx\) \(1.15050 - 0.567711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.780 + 1.17i)T \)
3 \( 1 - T \)
7 \( 1 + (2.56 - 0.662i)T \)
good5 \( 1 - 1.69iT - 5T^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + 6.04iT - 13T^{2} \)
17 \( 1 - 4.34iT - 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 3.02iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + 7.73iT - 41T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 9.43iT - 61T^{2} \)
67 \( 1 - 2.06iT - 67T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 - 3.39iT - 73T^{2} \)
79 \( 1 - 4.71iT - 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 - 7.73iT - 89T^{2} \)
97 \( 1 - 8.68iT - 97T^{2} \)
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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

−14.03082066160616251206646421165, −12.85070274972671156398235144536, −12.36794968127784047306013511253, −10.56506956350533307875725973254, −10.14149697683916050809631751607, −8.779536203045888251583713712097, −7.05163606503049647327694385645, −5.63654962368979661253963052247, −3.72559373255599200566254499478, −2.59189804351623496803671798464, 3.32663223976659181890032032957, 4.71809819236262540270338127616, 6.30541336418531402754248064819, 7.43576102295958274269696312276, 8.873064008314352261571920007980, 9.434632665027465740325450624038, 11.54881335786930881236294475521, 12.67490292108447914504051721000, 13.66393433576979398668238544688, 14.13812184330340239616088043459

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