Properties

Label 2-84-7.4-c5-0-6
Degree $2$
Conductor $84$
Sign $-0.553 + 0.832i$
Analytic cond. $13.4722$
Root an. cond. $3.67045$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 7.79i)3-s + (−23.0 − 39.9i)5-s + (112. + 64.8i)7-s + (−40.5 − 70.1i)9-s + (315. − 546. i)11-s − 1.07e3·13-s − 415.·15-s + (−80.5 + 139. i)17-s + (−588. − 1.01e3i)19-s + (1.01e3 − 583. i)21-s + (−1.08e3 − 1.87e3i)23-s + (499. − 864. i)25-s − 729·27-s − 4.49e3·29-s + (−159. + 275. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.412 − 0.714i)5-s + (0.866 + 0.500i)7-s + (−0.166 − 0.288i)9-s + (0.786 − 1.36i)11-s − 1.77·13-s − 0.476·15-s + (−0.0676 + 0.117i)17-s + (−0.373 − 0.647i)19-s + (0.500 − 0.288i)21-s + (−0.426 − 0.738i)23-s + (0.159 − 0.276i)25-s − 0.192·27-s − 0.991·29-s + (−0.0297 + 0.0515i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.553 + 0.832i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.709868 - 1.32480i\)
\(L(\frac12)\) \(\approx\) \(0.709868 - 1.32480i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 + (-112. - 64.8i)T \)
good5 \( 1 + (23.0 + 39.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-315. + 546. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + (80.5 - 139. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (588. + 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.08e3 + 1.87e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.49e3T + 2.05e7T^{2} \)
31 \( 1 + (159. - 275. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (7.59e3 + 1.31e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.05e4T + 1.15e8T^{2} \)
43 \( 1 + 455.T + 1.47e8T^{2} \)
47 \( 1 + (-1.03e4 - 1.79e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.65e3 + 1.67e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.18e3 - 5.51e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.45e4 - 4.25e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.70e4 - 2.94e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.29e4T + 1.80e9T^{2} \)
73 \( 1 + (-4.43e3 + 7.67e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.72e4 + 2.98e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.04e3T + 3.93e9T^{2} \)
89 \( 1 + (-1.01e4 - 1.75e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 5.40e4T + 8.58e9T^{2} \)
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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

−12.72890334177401793563532205446, −12.01495564874337591776574733795, −11.00711257016931531405157934910, −9.182087485484873618611260321011, −8.447653488393144275511144238186, −7.32389543320265269165928588100, −5.69917077417117844043269322620, −4.31797698229704536341713315901, −2.36035497088433317751405215993, −0.59708451868360033029805090933, 2.05863421924980949923692904651, 3.87601836887854677612210552990, 4.97142498976516659030371445022, 7.05698464223477029641445075135, 7.76275016492297043572347826177, 9.438423143405865236972338388396, 10.29674782473129323001954977895, 11.46719893314675671292244673145, 12.40306779388706635323732208957, 14.07675148872478654453447857927

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