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Properties

Label 2-84-21.17-c3-0-2
Degree $2$
Conductor $84$
Sign $0.967 - 0.253i$
Analytic cond. $4.95616$
Root an. cond. $2.22624$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.62 − 3.71i)3-s + (9.68 + 16.7i)5-s + (−2.66 + 18.3i)7-s + (−0.675 − 26.9i)9-s + (36.3 + 20.9i)11-s − 77.1i·13-s + (97.5 + 24.8i)15-s + (2.98 − 5.16i)17-s + (−59.0 + 34.0i)19-s + (58.4 + 76.4i)21-s + (30.8 − 17.7i)23-s + (−125. + 216. i)25-s + (−102. − 95.4i)27-s − 228. i·29-s + (62.3 + 36.0i)31-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)3-s + (0.866 + 1.50i)5-s + (−0.144 + 0.989i)7-s + (−0.0250 − 0.999i)9-s + (0.995 + 0.574i)11-s − 1.64i·13-s + (1.67 + 0.427i)15-s + (0.0425 − 0.0737i)17-s + (−0.712 + 0.411i)19-s + (0.607 + 0.794i)21-s + (0.279 − 0.161i)23-s + (−1.00 + 1.73i)25-s + (−0.733 − 0.680i)27-s − 1.46i·29-s + (0.361 + 0.208i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 0.967 - 0.253i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.07464 + 0.267512i\)
\(L(\frac12)\) \(\approx\) \(2.07464 + 0.267512i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.62 + 3.71i)T \)
7 \( 1 + (2.66 - 18.3i)T \)
good5 \( 1 + (-9.68 - 16.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.3 - 20.9i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 77.1iT - 2.19e3T^{2} \)
17 \( 1 + (-2.98 + 5.16i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (59.0 - 34.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-30.8 + 17.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 228. iT - 2.43e4T^{2} \)
31 \( 1 + (-62.3 - 36.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (34.6 + 60.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 132.T + 6.89e4T^{2} \)
43 \( 1 + 366.T + 7.95e4T^{2} \)
47 \( 1 + (-90.9 - 157. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-15.3 - 8.83i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (247. - 428. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-498. + 287. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-202. + 350. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 489. iT - 3.57e5T^{2} \)
73 \( 1 + (336. + 194. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (126. + 219. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 356.T + 5.71e5T^{2} \)
89 \( 1 + (-711. - 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 694. iT - 9.12e5T^{2} \)
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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

−13.90351500407372365694926880223, −12.87936479339631917345654077504, −11.82978507859443949064035550799, −10.35388813115857967913767894205, −9.418681443703186780103552770038, −8.064568752502636698344600889722, −6.73196741727484090663658924853, −5.93336144244423448048833448299, −3.21323809602125019798049308694, −2.13175717294791836549025413727, 1.55073863501661293591694524838, 3.94418663679176014540260452315, 4.93790157549624538543797228706, 6.69704762277297009720501648603, 8.589090597327426183897729848003, 9.139360496604115909972576226652, 10.11560112626631505799252278498, 11.51195829911761440534966120101, 13.02323341188999702297313052522, 13.78233854666605402709974979258

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