Properties

Label 2-84-7.4-c11-0-13
Degree $2$
Conductor $84$
Sign $-0.646 - 0.762i$
Analytic cond. $64.5408$
Root an. cond. $8.03373$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−121.5 + 210. i)3-s + (−5.53e3 − 9.59e3i)5-s + (−3.20e4 + 3.08e4i)7-s + (−2.95e4 − 5.11e4i)9-s + (4.51e5 − 7.82e5i)11-s − 7.61e5·13-s + 2.69e6·15-s + (2.39e6 − 4.14e6i)17-s + (−2.25e6 − 3.90e6i)19-s + (−2.60e6 − 1.04e7i)21-s + (−2.38e7 − 4.13e7i)23-s + (−3.69e7 + 6.40e7i)25-s + 1.43e7·27-s + 1.65e8·29-s + (−2.95e7 + 5.11e7i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.792 − 1.37i)5-s + (−0.720 + 0.693i)7-s + (−0.166 − 0.288i)9-s + (0.845 − 1.46i)11-s − 0.568·13-s + 0.915·15-s + (0.408 − 0.707i)17-s + (−0.208 − 0.361i)19-s + (−0.139 − 0.560i)21-s + (−0.773 − 1.34i)23-s + (−0.757 + 1.31i)25-s + 0.192·27-s + 1.49·29-s + (−0.185 + 0.321i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $-0.646 - 0.762i$
Analytic conductor: \(64.5408\)
Root analytic conductor: \(8.03373\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{84} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :11/2),\ -0.646 - 0.762i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0390225 + 0.0842711i\)
\(L(\frac12)\) \(\approx\) \(0.0390225 + 0.0842711i\)
\(L(\frac{13}{2})\) 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 \( 1 + (121.5 - 210. i)T \)
7 \( 1 + (3.20e4 - 3.08e4i)T \)
good5 \( 1 + (5.53e3 + 9.59e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-4.51e5 + 7.82e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 7.61e5T + 1.79e12T^{2} \)
17 \( 1 + (-2.39e6 + 4.14e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (2.25e6 + 3.90e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (2.38e7 + 4.13e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 1.65e8T + 1.22e16T^{2} \)
31 \( 1 + (2.95e7 - 5.11e7i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-1.08e8 - 1.87e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 5.52e8T + 5.50e17T^{2} \)
43 \( 1 + 1.04e9T + 9.29e17T^{2} \)
47 \( 1 + (6.93e7 + 1.20e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-2.34e9 + 4.05e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (4.49e9 - 7.78e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (1.08e8 + 1.87e8i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (8.64e9 - 1.49e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 5.28e9T + 2.31e20T^{2} \)
73 \( 1 + (5.55e9 - 9.61e9i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (-2.14e9 - 3.71e9i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 4.61e10T + 1.28e21T^{2} \)
89 \( 1 + (8.53e9 + 1.47e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 1.57e11T + 7.15e21T^{2} \)
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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

−11.77537799850268108589636053366, −10.12084767475639802746015273936, −8.896812247846353668563122871487, −8.413014284073783252813584337965, −6.52857676431261970857254628821, −5.32177364212949523524443919114, −4.26582934555001369584350333958, −3.00957096044531033836221040057, −0.872300064039292173025284101546, −0.03120609512125453469326498569, 1.71079973057188848559060575103, 3.22544646253221817548704109614, 4.26941234289570533492608424557, 6.27926498650771006932834797517, 7.08027414773079277068989184285, 7.76813511680059574706158630837, 9.768554607065371128601772486398, 10.50945501019021903601445158615, 11.78025907209263415315664793238, 12.44154615634166519508798650361

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