↑

Properties

Label 2-84-7.2-c5-0-2
Degree $2$
Conductor $84$
Sign $0.967 + 0.252i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 − 7.79i)3-s + (−32.7 + 56.7i)5-s + (40.6 − 123. i)7-s + (−40.5 + 70.1i)9-s + (122. + 212. i)11-s + 434.·13-s + 590.·15-s + (551. + 954. i)17-s + (1.43e3 − 2.49e3i)19-s + (−1.14e3 + 237. i)21-s + (2.11e3 − 3.66e3i)23-s + (−587. − 1.01e3i)25-s + 729·27-s + 4.96e3·29-s + (4.39e3 + 7.60e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.586 + 1.01i)5-s + (0.313 − 0.949i)7-s + (−0.166 + 0.288i)9-s + (0.305 + 0.529i)11-s + 0.713·13-s + 0.677·15-s + (0.462 + 0.801i)17-s + (0.914 − 1.58i)19-s + (−0.565 + 0.117i)21-s + (0.833 − 1.44i)23-s + (−0.187 − 0.325i)25-s + 0.192·27-s + 1.09·29-s + (0.820 + 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $0.967 + 0.252i$
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: \((2,\ 84,\ (\ :5/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.56939 - 0.201627i\)
\(L(\frac12)\) \(\approx\) \(1.56939 - 0.201627i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-40.6 + 123. i)T \)
good5 \( 1 + (32.7 - 56.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-122. - 212. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 434.T + 3.71e5T^{2} \)
17 \( 1 + (-551. - 954. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.43e3 + 2.49e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.11e3 + 3.66e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 4.96e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.39e3 - 7.60e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.22e3 + 2.11e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 3.66e3T + 1.15e8T^{2} \)
43 \( 1 + 7.19e3T + 1.47e8T^{2} \)
47 \( 1 + (1.63e3 - 2.83e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.51e3 - 2.61e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.57e4 - 4.45e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-6.65e3 + 1.15e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.54e4 + 2.67e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.18e4T + 1.80e9T^{2} \)
73 \( 1 + (1.73e4 + 2.99e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.87e4 - 6.71e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.00e5T + 3.93e9T^{2} \)
89 \( 1 + (2.03e4 - 3.53e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.40e5T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.33064502462539439191746420152, −12.05291833619398836171219747383, −11.02842325538722746114786060879, −10.34928873525461886716739690646, −8.509879180809295599831043552427, −7.21260997944316618923009705267, −6.62331135394178576187184158284, −4.64343726847038328327050505178, −3.09244846324587534226707969096, −0.990555564616122450622168838021, 1.05633571407992533871828736756, 3.46041546340850499760503935043, 4.94664991579940228256524848705, 5.91514265393052544770492906593, 7.930966958132089606405990448490, 8.830356180626795159502430988931, 9.886892997296700479002632918971, 11.69238872535378780076750628609, 11.79122298824871458356527287760, 13.26027166012384193637313671824

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