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Properties

Label 2-84-7.5-c8-0-7
Degree $2$
Conductor $84$
Sign $-0.577 + 0.816i$
Analytic cond. $34.2198$
Root an. cond. $5.84976$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 + 23.3i)3-s + (−374. − 216. i)5-s + (83.7 + 2.39e3i)7-s + (1.09e3 − 1.89e3i)9-s + (6.90e3 + 1.19e4i)11-s + 3.22e3i·13-s + 2.02e4·15-s + (−2.13e4 + 1.23e4i)17-s + (2.91e4 + 1.68e4i)19-s + (−5.95e4 − 9.52e4i)21-s + (−1.12e5 + 1.95e5i)23-s + (−1.01e5 − 1.76e5i)25-s + 1.02e5i·27-s − 9.08e5·29-s + (9.25e5 − 5.34e5i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.598 − 0.345i)5-s + (0.0348 + 0.999i)7-s + (0.166 − 0.288i)9-s + (0.471 + 0.816i)11-s + 0.112i·13-s + 0.399·15-s + (−0.255 + 0.147i)17-s + (0.223 + 0.129i)19-s + (−0.305 − 0.489i)21-s + (−0.403 + 0.698i)23-s + (−0.260 − 0.451i)25-s + 0.192i·27-s − 1.28·29-s + (1.00 − 0.578i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(34.2198\)
Root analytic conductor: \(5.84976\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :4),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0875281 - 0.169086i\)
\(L(\frac12)\) \(\approx\) \(0.0875281 - 0.169086i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (40.5 - 23.3i)T \)
7 \( 1 + (-83.7 - 2.39e3i)T \)
good5 \( 1 + (374. + 216. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-6.90e3 - 1.19e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.22e3iT - 8.15e8T^{2} \)
17 \( 1 + (2.13e4 - 1.23e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-2.91e4 - 1.68e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.12e5 - 1.95e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 9.08e5T + 5.00e11T^{2} \)
31 \( 1 + (-9.25e5 + 5.34e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-9.79e5 + 1.69e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 2.90e6iT - 7.98e12T^{2} \)
43 \( 1 + 4.01e6T + 1.16e13T^{2} \)
47 \( 1 + (4.29e6 + 2.47e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (3.98e6 + 6.90e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-4.69e6 + 2.71e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.84e6 - 1.06e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (9.53e6 + 1.65e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 2.08e7T + 6.45e14T^{2} \)
73 \( 1 + (9.66e6 - 5.57e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-1.21e7 + 2.10e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 3.92e7iT - 2.25e15T^{2} \)
89 \( 1 + (-1.02e7 - 5.89e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 7.80e7iT - 7.83e15T^{2} \)
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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

−12.00377527934211090057091920439, −11.50866228523544092183502040965, −9.964603545279901473674605407994, −8.987591164572935285627218883159, −7.71667066353022645488038057714, −6.25926950053144897977506807955, −5.03266005287625550452132458722, −3.82432635565497574688217181613, −1.92240591224259681457900327343, −0.06703420110919159801837258143, 1.20098374809339338187612843535, 3.27933528204873517908782461409, 4.55350361678070674969254073023, 6.19880723243701836594771972649, 7.22634812954792206984433126180, 8.287345578209084785382184226288, 9.910185162795576694889247401231, 11.07718781665636377281298788838, 11.66300050751165326652952762233, 13.05837809334015760349413112875

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