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Properties

Label 2-84-12.11-c5-0-0
Degree $2$
Conductor $84$
Sign $-0.000383 + 0.999i$
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  + (1.70 + 5.39i)2-s + (−12.7 − 8.95i)3-s + (−26.2 + 18.3i)4-s + 95.2i·5-s + (26.5 − 84.0i)6-s − 49i·7-s + (−143. − 110. i)8-s + (82.6 + 228. i)9-s + (−513. + 162. i)10-s − 125.·11-s + (498. + 0.191i)12-s − 253.·13-s + (264. − 83.4i)14-s + (852. − 1.21e3i)15-s + (349. − 962. i)16-s − 1.11e3i·17-s + ⋯
L(s)  = 1  + (0.301 + 0.953i)2-s + (−0.818 − 0.574i)3-s + (−0.818 + 0.574i)4-s + 1.70i·5-s + (0.301 − 0.953i)6-s − 0.377i·7-s + (−0.793 − 0.608i)8-s + (0.340 + 0.940i)9-s + (−1.62 + 0.512i)10-s − 0.313·11-s + (0.999 + 0.000383i)12-s − 0.416·13-s + (0.360 − 0.113i)14-s + (0.978 − 1.39i)15-s + (0.340 − 0.940i)16-s − 0.934i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000383 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.000383 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.000383 + 0.999i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.000383 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00723078 - 0.00723356i\)
\(L(\frac12)\) \(\approx\) \(0.00723078 - 0.00723356i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.70 - 5.39i)T \)
3 \( 1 + (12.7 + 8.95i)T \)
7 \( 1 + 49iT \)
good5 \( 1 - 95.2iT - 3.12e3T^{2} \)
11 \( 1 + 125.T + 1.61e5T^{2} \)
13 \( 1 + 253.T + 3.71e5T^{2} \)
17 \( 1 + 1.11e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.55e3iT - 2.47e6T^{2} \)
23 \( 1 + 237.T + 6.43e6T^{2} \)
29 \( 1 + 7.32e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.61e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.78e3T + 6.93e7T^{2} \)
41 \( 1 - 1.81e4iT - 1.15e8T^{2} \)
43 \( 1 + 6.77e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.99e4T + 2.29e8T^{2} \)
53 \( 1 - 1.89e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.32e4T + 7.14e8T^{2} \)
61 \( 1 - 1.23e4T + 8.44e8T^{2} \)
67 \( 1 + 4.92e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.10e4T + 1.80e9T^{2} \)
73 \( 1 + 4.59e4T + 2.07e9T^{2} \)
79 \( 1 + 8.17e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.75e4T + 3.93e9T^{2} \)
89 \( 1 - 1.20e5iT - 5.58e9T^{2} \)
97 \( 1 - 2.52e4T + 8.58e9T^{2} \)
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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

−14.03790671868331933255310383687, −13.38659272703216485445499013470, −11.97351508593036836629400822396, −10.96310339485530394306753575760, −9.819511335021466091694135506572, −7.78857630081930086528600037356, −7.01394550726800607019692172967, −6.26623653535599467586553638564, −4.80318401987669138133816759708, −2.88963713775046128894864088308, 0.00439549853971399165938518163, 1.50327736251664077892305301446, 3.91005136705617839747148647531, 5.01690161788142947800063626378, 5.79280058389726227413694622500, 8.403881803960862713021142643491, 9.385977383994689167093637818319, 10.33517335300376506811932269648, 11.60226593514410812409214860360, 12.50070633752654277509273000995

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