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Properties

Label 2-84-7.4-c5-0-4
Degree $2$
Conductor $84$
Sign $0.706 + 0.707i$
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 + (15.9 + 27.6i)5-s + (85.7 − 97.2i)7-s + (−40.5 − 70.1i)9-s + (−130. + 225. i)11-s + 769.·13-s + 287.·15-s + (776. − 1.34e3i)17-s + (−375. − 649. i)19-s + (−372. − 1.10e3i)21-s + (−377. − 653. i)23-s + (1.05e3 − 1.82e3i)25-s − 729·27-s + 6.00e3·29-s + (3.21e3 − 5.55e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.285 + 0.495i)5-s + (0.661 − 0.750i)7-s + (−0.166 − 0.288i)9-s + (−0.325 + 0.562i)11-s + 1.26·13-s + 0.330·15-s + (0.651 − 1.12i)17-s + (−0.238 − 0.412i)19-s + (−0.184 − 0.547i)21-s + (−0.148 − 0.257i)23-s + (0.336 − 0.582i)25-s − 0.192·27-s + 1.32·29-s + (0.599 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.706 + 0.707i$
Analytic conductor: \(13.4722\)
Root analytic conductor: \(3.67045\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :5/2),\ 0.706 + 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.08237 - 0.862844i\)
\(L(\frac12)\) \(\approx\) \(2.08237 - 0.862844i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-85.7 + 97.2i)T \)
good5 \( 1 + (-15.9 - 27.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (130. - 225. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 769.T + 3.71e5T^{2} \)
17 \( 1 + (-776. + 1.34e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (375. + 649. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (377. + 653. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 6.00e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.21e3 + 5.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-2.38e3 - 4.13e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 5.42e3T + 1.15e8T^{2} \)
43 \( 1 + 1.18e4T + 1.47e8T^{2} \)
47 \( 1 + (-8.71e3 - 1.50e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.88e4 - 3.26e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.10e4 - 1.91e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-4.08e3 - 7.07e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (6.50e3 - 1.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.23e4T + 1.80e9T^{2} \)
73 \( 1 + (2.18e4 - 3.77e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.83e4 + 6.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 2.18e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.84e4 - 1.18e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 9.30e4T + 8.58e9T^{2} \)
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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

−13.49053382475129142357458678149, −12.08277487345164215345836449691, −10.96174922226702206192855065815, −9.956867790221952783321740832168, −8.450270798764431032088715076058, −7.40914074779058748453269482407, −6.29207697722968644524151047594, −4.54282996525474825334978965668, −2.77977641546340569007566674488, −1.08328569138545676589532022316, 1.53819516409194577005221787439, 3.41957957351804240929443631520, 5.03404498602410703001881730994, 6.10208303215763370207627473459, 8.269299887435917048780181716560, 8.675183927893179770482470021399, 10.14607842283553332231022722423, 11.16912294207430047356182819325, 12.38804560447090398750272309344, 13.51605742690924285867595647623

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