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Properties

Label 2-84-7.2-c5-0-0
Degree $2$
Conductor $84$
Sign $-0.236 - 0.971i$
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 + (9.28 − 16.0i)5-s + (−127. + 22.6i)7-s + (−40.5 + 70.1i)9-s + (80.7 + 139. i)11-s + 14.1·13-s − 167.·15-s + (382. + 663. i)17-s + (−707. + 1.22e3i)19-s + (750. + 893. i)21-s + (−2.09e3 + 3.62e3i)23-s + (1.38e3 + 2.40e3i)25-s + 729·27-s − 4.20e3·29-s + (1.19e3 + 2.06e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.166 − 0.287i)5-s + (−0.984 + 0.174i)7-s + (−0.166 + 0.288i)9-s + (0.201 + 0.348i)11-s + 0.0231·13-s − 0.191·15-s + (0.321 + 0.556i)17-s + (−0.449 + 0.778i)19-s + (0.371 + 0.441i)21-s + (−0.824 + 1.42i)23-s + (0.444 + 0.770i)25-s + 0.192·27-s − 0.927·29-s + (0.223 + 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.236 - 0.971i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ -0.236 - 0.971i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.420563 + 0.535315i\)
\(L(\frac12)\) \(\approx\) \(0.420563 + 0.535315i\)
\(L(\frac{7}{2})\) 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 \( 1 + (127. - 22.6i)T \)
good5 \( 1 + (-9.28 + 16.0i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-80.7 - 139. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 14.1T + 3.71e5T^{2} \)
17 \( 1 + (-382. - 663. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (707. - 1.22e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (2.09e3 - 3.62e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 4.20e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.19e3 - 2.06e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-336. + 582. i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 4.17e3T + 1.15e8T^{2} \)
43 \( 1 + 5.43e3T + 1.47e8T^{2} \)
47 \( 1 + (3.15e3 - 5.45e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (8.20e3 + 1.42e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.98e3 + 3.43e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.51e4 + 4.35e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (6.82e3 + 1.18e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 8.39e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.42e4 - 2.47e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.99e4 + 5.19e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 6.15e4T + 3.93e9T^{2} \)
89 \( 1 + (2.11e4 - 3.66e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.46e4T + 8.58e9T^{2} \)
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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

−13.31941268520593710726393287262, −12.64108948737351152459860660333, −11.65755482314712753275476612338, −10.23529982403254079026505839229, −9.229906250379144721349727138087, −7.82317852839614520897010367604, −6.53547292710622975556494647681, −5.48120750112419088273079071298, −3.58947779920033430569995968377, −1.66232514891216347565067728470, 0.28835352411922524929152281949, 2.82299099144683285795690919074, 4.29402426594618259775622973822, 5.92918858536184659044975163765, 6.92978397891735721025252232725, 8.657408210756200445601072456362, 9.814424209138384949204326472181, 10.64471633889285557328115468650, 11.85755440863421857066886354061, 12.99351674760848623853549496902

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