↑

Properties

Label 2-2e3-8.3-c14-0-10
Degree $2$
Conductor $8$
Sign $-0.397 + 0.917i$
Analytic cond. $9.94631$
Root an. cond. $3.15377$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (118. − 48.3i)2-s − 1.15e3·3-s + (1.17e4 − 1.14e4i)4-s + 9.66e3i·5-s + (−1.36e5 + 5.57e4i)6-s − 1.34e6i·7-s + (8.34e5 − 1.92e6i)8-s − 3.45e6·9-s + (4.67e5 + 1.14e6i)10-s − 8.97e5·11-s + (−1.35e7 + 1.32e7i)12-s − 5.37e7i·13-s + (−6.50e7 − 1.59e8i)14-s − 1.11e7i·15-s + (5.91e6 − 2.68e8i)16-s − 1.06e8·17-s + ⋯
L(s)  = 1  + (0.925 − 0.377i)2-s − 0.527·3-s + (0.714 − 0.699i)4-s + 0.123i·5-s + (−0.488 + 0.199i)6-s − 1.63i·7-s + (0.397 − 0.917i)8-s − 0.722·9-s + (0.0467 + 0.114i)10-s − 0.0460·11-s + (−0.376 + 0.368i)12-s − 0.857i·13-s + (−0.617 − 1.51i)14-s − 0.0652i·15-s + (0.0220 − 0.999i)16-s − 0.259·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.397 + 0.917i$
Analytic conductor: \(9.94631\)
Root analytic conductor: \(3.15377\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :7),\ -0.397 + 0.917i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.20008 - 1.82856i\)
\(L(\frac12)\) \(\approx\) \(1.20008 - 1.82856i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-118. + 48.3i)T \)
good3 \( 1 + 1.15e3T + 4.78e6T^{2} \)
5 \( 1 - 9.66e3iT - 6.10e9T^{2} \)
7 \( 1 + 1.34e6iT - 6.78e11T^{2} \)
11 \( 1 + 8.97e5T + 3.79e14T^{2} \)
13 \( 1 + 5.37e7iT - 3.93e15T^{2} \)
17 \( 1 + 1.06e8T + 1.68e17T^{2} \)
19 \( 1 - 9.74e8T + 7.99e17T^{2} \)
23 \( 1 - 4.91e9iT - 1.15e19T^{2} \)
29 \( 1 - 1.32e10iT - 2.97e20T^{2} \)
31 \( 1 - 3.03e8iT - 7.56e20T^{2} \)
37 \( 1 + 7.87e10iT - 9.01e21T^{2} \)
41 \( 1 + 2.80e11T + 3.79e22T^{2} \)
43 \( 1 - 3.24e11T + 7.38e22T^{2} \)
47 \( 1 + 2.85e10iT - 2.56e23T^{2} \)
53 \( 1 + 2.31e12iT - 1.37e24T^{2} \)
59 \( 1 - 3.41e12T + 6.19e24T^{2} \)
61 \( 1 - 5.23e12iT - 9.87e24T^{2} \)
67 \( 1 - 7.72e12T + 3.67e25T^{2} \)
71 \( 1 - 5.57e12iT - 8.27e25T^{2} \)
73 \( 1 + 4.34e12T + 1.22e26T^{2} \)
79 \( 1 + 2.63e13iT - 3.68e26T^{2} \)
83 \( 1 + 3.06e13T + 7.36e26T^{2} \)
89 \( 1 - 3.63e13T + 1.95e27T^{2} \)
97 \( 1 - 7.63e13T + 6.52e27T^{2} \)
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

−17.61004198531031113544821687864, −16.20005071241594507104192406045, −14.38739240479442914335663716542, −13.20485072906447241996961565804, −11.43382959858815767026710715588, −10.33178675497708328185770457691, −7.14809449477386439827619910232, −5.30459525712283686568098075219, −3.42678150360528532137602526213, −0.841414775344892205727968347331, 2.60343461002363547212616341344, 5.05241322465936846080236336931, 6.32500906862010424548057312458, 8.658661823672135355255472661365, 11.46636737583749215302669191003, 12.38854563739458707126510968958, 14.25894371617250801501947442881, 15.62360604749843609362959518916, 16.88232333497473952022432229268, 18.53079542247788088298427525239

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