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Properties

Label 2-84-7.2-c5-0-4
Degree $2$
Conductor $84$
Sign $-0.580 + 0.814i$
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 + (−39.3 + 68.1i)5-s + (−100. − 81.7i)7-s + (−40.5 + 70.1i)9-s + (−345. − 598. i)11-s + 818.·13-s − 708.·15-s + (−554. − 960. i)17-s + (286. − 496. i)19-s + (184. − 1.15e3i)21-s + (−1.25e3 + 2.18e3i)23-s + (−1.53e3 − 2.65e3i)25-s − 729·27-s − 3.25e3·29-s + (−5.05e3 − 8.76e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.703 + 1.21i)5-s + (−0.776 − 0.630i)7-s + (−0.166 + 0.288i)9-s + (−0.861 − 1.49i)11-s + 1.34·13-s − 0.812·15-s + (−0.465 − 0.806i)17-s + (0.182 − 0.315i)19-s + (0.0913 − 0.570i)21-s + (−0.496 + 0.859i)23-s + (−0.490 − 0.849i)25-s − 0.192·27-s − 0.719·29-s + (−0.945 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.580 + 0.814i$
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: \((2,\ 84,\ (\ :5/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.114845 - 0.222843i\)
\(L(\frac12)\) \(\approx\) \(0.114845 - 0.222843i\)
\(L(\frac{7}{2})\) 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 \( 1 + (100. + 81.7i)T \)
good5 \( 1 + (39.3 - 68.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (345. + 598. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 818.T + 3.71e5T^{2} \)
17 \( 1 + (554. + 960. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-286. + 496. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.25e3 - 2.18e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.25e3T + 2.05e7T^{2} \)
31 \( 1 + (5.05e3 + 8.76e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.43e3 - 4.21e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + 9.30e3T + 1.47e8T^{2} \)
47 \( 1 + (6.45e3 - 1.11e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-9.77e3 - 1.69e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.25e4 - 2.17e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.56e4 + 2.71e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.79e4 + 4.84e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 2.05e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.38e4 - 5.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (7.03e3 - 1.21e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 7.71e4T + 3.93e9T^{2} \)
89 \( 1 + (160. - 277. i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.12e5T + 8.58e9T^{2} \)
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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

−13.39730578106393180371757284254, −11.25379554717136408593980789242, −10.99313555177634479167353052233, −9.720468430261817521506583853813, −8.306685128970502208157027819600, −7.16485187391012257037199814276, −5.86943257441109468115923548512, −3.75439062531395380934642240308, −3.06782495035286905844898922359, −0.097173184674663072015220257702, 1.80678864141657138763027891625, 3.76456480697270000393272163943, 5.27893519718277550736754900197, 6.80461502144275504171245742293, 8.232673749181011799991607386553, 8.884804332923566815831229099069, 10.29127135283901479309559263535, 11.93095783058076930191312645132, 12.73647252415185237504347867009, 13.17664300971680886733399317619

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