Properties

Label 2-6-3.2-c24-0-7
Degree $2$
Conductor $6$
Sign $-0.861 - 0.507i$
Analytic cond. $21.8980$
Root an. cond. $4.67953$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.89e3i·2-s + (2.69e5 − 4.58e5i)3-s − 8.38e6·4-s − 4.56e8i·5-s + (−1.32e9 − 7.80e8i)6-s + 1.81e9·7-s + 2.42e10i·8-s + (−1.37e11 − 2.46e11i)9-s − 1.32e12·10-s − 3.36e12i·11-s + (−2.26e12 + 3.84e12i)12-s + 3.10e13·13-s − 5.26e12i·14-s + (−2.09e14 − 1.23e14i)15-s + 7.03e13·16-s + 1.14e14i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.507 − 0.861i)3-s − 0.500·4-s − 1.87i·5-s + (−0.609 − 0.358i)6-s + 0.131·7-s + 0.353i·8-s + (−0.485 − 0.874i)9-s − 1.32·10-s − 1.07i·11-s + (−0.253 + 0.430i)12-s + 1.33·13-s − 0.0929i·14-s + (−1.61 − 0.948i)15-s + 0.250·16-s + 0.196i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.861 - 0.507i$
Analytic conductor: \(21.8980\)
Root analytic conductor: \(4.67953\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :12),\ -0.861 - 0.507i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.538135 + 1.97529i\)
\(L(\frac12)\) \(\approx\) \(0.538135 + 1.97529i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.89e3iT \)
3 \( 1 + (-2.69e5 + 4.58e5i)T \)
good5 \( 1 + 4.56e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.81e9T + 1.91e20T^{2} \)
11 \( 1 + 3.36e12iT - 9.84e24T^{2} \)
13 \( 1 - 3.10e13T + 5.42e26T^{2} \)
17 \( 1 - 1.14e14iT - 3.39e29T^{2} \)
19 \( 1 - 9.25e14T + 4.89e30T^{2} \)
23 \( 1 - 2.51e16iT - 4.80e32T^{2} \)
29 \( 1 - 5.38e17iT - 1.25e35T^{2} \)
31 \( 1 - 4.67e17T + 6.20e35T^{2} \)
37 \( 1 - 3.11e18T + 4.33e37T^{2} \)
41 \( 1 + 1.15e19iT - 5.09e38T^{2} \)
43 \( 1 - 5.41e19T + 1.59e39T^{2} \)
47 \( 1 + 1.41e20iT - 1.35e40T^{2} \)
53 \( 1 + 5.85e20iT - 2.41e41T^{2} \)
59 \( 1 - 1.22e21iT - 3.16e42T^{2} \)
61 \( 1 - 3.16e21T + 7.04e42T^{2} \)
67 \( 1 + 2.88e21T + 6.69e43T^{2} \)
71 \( 1 - 5.86e21iT - 2.69e44T^{2} \)
73 \( 1 + 4.89e21T + 5.24e44T^{2} \)
79 \( 1 - 4.71e22T + 3.49e45T^{2} \)
83 \( 1 + 1.25e23iT - 1.14e46T^{2} \)
89 \( 1 + 4.49e22iT - 6.10e46T^{2} \)
97 \( 1 + 2.08e23T + 4.81e47T^{2} \)
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

−16.19767160223226297864499890177, −13.72662970065015801326559113315, −12.91246783363889159334080220727, −11.58874032772620171946499107270, −9.052172411390951683954085771110, −8.258892363732788380560713231672, −5.58259360166925018181375303861, −3.61993267637823713923120818443, −1.49601786135068435117804322497, −0.74175779133109537033970377374, 2.65544190318909419815519909205, 4.13828093260468041706972791137, 6.29074672298160608414819821961, 7.82285746095841308360086192322, 9.764307860287863220529750378022, 11.00243324681180410159624225387, 13.87533736879188994656877329385, 14.86393006651563751247063730639, 15.79642532688980066485844740093, 17.77543014573149285901999214131

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