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Properties

Label 2-84-7.4-c3-0-3
Degree $2$
Conductor $84$
Sign $-0.527 + 0.849i$
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  + (1.5 − 2.59i)3-s + (−4.91 − 8.50i)5-s + (−16.3 − 8.74i)7-s + (−4.5 − 7.79i)9-s + (7.08 − 12.2i)11-s − 26.1·13-s − 29.4·15-s + (39.2 − 68.0i)17-s + (−36.5 − 63.3i)19-s + (−47.2 + 29.2i)21-s + (48 + 83.1i)23-s + (14.2 − 24.6i)25-s − 27·27-s + 173.·29-s + (−33.6 + 58.2i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.439 − 0.761i)5-s + (−0.881 − 0.472i)7-s + (−0.166 − 0.288i)9-s + (0.194 − 0.336i)11-s − 0.557·13-s − 0.507·15-s + (0.560 − 0.971i)17-s + (−0.441 − 0.765i)19-s + (−0.490 + 0.304i)21-s + (0.435 + 0.753i)23-s + (0.113 − 0.197i)25-s − 0.192·27-s + 1.10·29-s + (−0.194 + 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ -0.527 + 0.849i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.557432 - 1.00177i\)
\(L(\frac12)\) \(\approx\) \(0.557432 - 1.00177i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (16.3 + 8.74i)T \)
good5 \( 1 + (4.91 + 8.50i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.08 + 12.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 26.1T + 2.19e3T^{2} \)
17 \( 1 + (-39.2 + 68.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (36.5 + 63.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48 - 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 + (33.6 - 58.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. - 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 472.T + 6.89e4T^{2} \)
43 \( 1 + 463.T + 7.95e4T^{2} \)
47 \( 1 + (-45.5 - 78.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-81.6 + 141. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-300. + 520. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (285. + 495. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-269. + 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (-221. + 383. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-22.8 - 39.5i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 686.T + 5.71e5T^{2} \)
89 \( 1 + (330. + 571. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 658.T + 9.12e5T^{2} \)
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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

−13.30177007856094058250977685806, −12.49066990568491455426125559000, −11.48129422008524655688178624662, −9.903057498267390880780454034494, −8.878496162589334890971248536935, −7.64725174898921561250186042742, −6.52674765005095420558009850269, −4.76956288459268093062528512620, −3.08858726264502574249966638880, −0.68426468633149209831045886963, 2.75070442573841578542245323906, 4.07236782090922010491321613640, 5.93316820101530037023887603126, 7.23113562034042865248576463590, 8.601523118034197292544683707033, 9.849679006616358412002857782343, 10.66563070725423610220240504987, 12.04753149751472493880809761521, 12.96641778209182791853576527243, 14.59353375711356349416584337107

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