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Properties

Label 2-84-7.4-c3-0-0
Degree $2$
Conductor $84$
Sign $-0.699 - 0.714i$
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 + (0.723 + 1.25i)5-s + (−12.3 + 13.7i)7-s + (−4.5 − 7.79i)9-s + (−23.0 + 39.9i)11-s − 32.2·13-s − 4.33·15-s + (−38.8 + 67.3i)17-s + (−6.33 − 10.9i)19-s + (−17.1 − 52.8i)21-s + (50.4 + 87.4i)23-s + (61.4 − 106. i)25-s + 27·27-s + 213.·29-s + (−21.0 + 36.4i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.0646 + 0.112i)5-s + (−0.669 + 0.743i)7-s + (−0.166 − 0.288i)9-s + (−0.632 + 1.09i)11-s − 0.687·13-s − 0.0746·15-s + (−0.554 + 0.961i)17-s + (−0.0764 − 0.132i)19-s + (−0.178 − 0.549i)21-s + (0.457 + 0.792i)23-s + (0.491 − 0.851i)25-s + 0.192·27-s + 1.36·29-s + (−0.121 + 0.211i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.699 - 0.714i$
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: \((2,\ 84,\ (\ :3/2),\ -0.699 - 0.714i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.328265 + 0.780321i\)
\(L(\frac12)\) \(\approx\) \(0.328265 + 0.780321i\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.3 - 13.7i)T \)
good5 \( 1 + (-0.723 - 1.25i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (23.0 - 39.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.2T + 2.19e3T^{2} \)
17 \( 1 + (38.8 - 67.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6.33 + 10.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-50.4 - 87.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 213.T + 2.43e4T^{2} \)
31 \( 1 + (21.0 - 36.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (155. + 268. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 44.0T + 6.89e4T^{2} \)
43 \( 1 - 381.T + 7.95e4T^{2} \)
47 \( 1 + (-179. - 310. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (92.4 - 160. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (227. - 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (5.92 + 10.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (295. - 511. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 494.T + 3.57e5T^{2} \)
73 \( 1 + (487. - 844. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (149. + 259. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + (-347. - 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 481.T + 9.12e5T^{2} \)
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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

−14.36678092739282170844435708009, −12.80735551457251238395072572961, −12.22063700813196695341483214600, −10.73542001213801985502443094936, −9.853226891659297405119168916251, −8.780437512202873163813447330148, −7.15704491940414772297050927207, −5.82313109635435076567666520079, −4.48912220867212655707010542272, −2.59954412378196091578593228608, 0.51754181997242148269473396959, 2.94545255976121480248048515171, 4.91413425486300041750861112943, 6.40668805022569083674497547561, 7.44595059800541147959507452981, 8.816554730569773242426660490544, 10.21105947490999669918708061280, 11.17786635934676045711515401140, 12.44194284040330480511345818280, 13.37823648090706507954903407363

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