↑

Properties

Label 2-2e3-8.5-c19-0-3
Degree $2$
Conductor $8$
Sign $0.206 + 0.978i$
Analytic cond. $18.3053$
Root an. cond. $4.27847$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−600. + 404. i)2-s + 6.46e4i·3-s + (1.96e5 − 4.85e5i)4-s + 4.52e6i·5-s + (−2.61e7 − 3.88e7i)6-s − 6.45e7·7-s + (7.84e7 + 3.71e8i)8-s − 3.01e9·9-s + (−1.82e9 − 2.71e9i)10-s + 9.68e9i·11-s + (3.14e10 + 1.27e10i)12-s + 7.19e9i·13-s + (3.87e10 − 2.61e10i)14-s − 2.92e11·15-s + (−1.97e11 − 1.91e11i)16-s − 9.85e10·17-s + ⋯
L(s)  = 1  + (−0.829 + 0.558i)2-s + 1.89i·3-s + (0.375 − 0.926i)4-s + 1.03i·5-s + (−1.05 − 1.57i)6-s − 0.604·7-s + (0.206 + 0.978i)8-s − 2.59·9-s + (−0.578 − 0.858i)10-s + 1.23i·11-s + (1.75 + 0.711i)12-s + 0.188i·13-s + (0.501 − 0.337i)14-s − 1.96·15-s + (−0.718 − 0.695i)16-s − 0.201·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.206 + 0.978i$
Analytic conductor: \(18.3053\)
Root analytic conductor: \(4.27847\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :19/2),\ 0.206 + 0.978i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.542762 - 0.440126i\)
\(L(\frac12)\) \(\approx\) \(0.542762 - 0.440126i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (600. - 404. i)T \)
good3 \( 1 - 6.46e4iT - 1.16e9T^{2} \)
5 \( 1 - 4.52e6iT - 1.90e13T^{2} \)
7 \( 1 + 6.45e7T + 1.13e16T^{2} \)
11 \( 1 - 9.68e9iT - 6.11e19T^{2} \)
13 \( 1 - 7.19e9iT - 1.46e21T^{2} \)
17 \( 1 + 9.85e10T + 2.39e23T^{2} \)
19 \( 1 + 5.79e11iT - 1.97e24T^{2} \)
23 \( 1 - 1.28e13T + 7.46e25T^{2} \)
29 \( 1 - 1.42e14iT - 6.10e27T^{2} \)
31 \( 1 - 1.10e14T + 2.16e28T^{2} \)
37 \( 1 - 4.03e14iT - 6.24e29T^{2} \)
41 \( 1 - 2.24e15T + 4.39e30T^{2} \)
43 \( 1 + 2.18e15iT - 1.08e31T^{2} \)
47 \( 1 + 6.50e15T + 5.88e31T^{2} \)
53 \( 1 + 1.93e16iT - 5.77e32T^{2} \)
59 \( 1 + 1.98e16iT - 4.42e33T^{2} \)
61 \( 1 - 5.65e16iT - 8.34e33T^{2} \)
67 \( 1 + 9.83e16iT - 4.95e34T^{2} \)
71 \( 1 + 6.44e17T + 1.49e35T^{2} \)
73 \( 1 + 1.55e16T + 2.53e35T^{2} \)
79 \( 1 - 1.06e18T + 1.13e36T^{2} \)
83 \( 1 - 1.05e18iT - 2.90e36T^{2} \)
89 \( 1 + 1.47e16T + 1.09e37T^{2} \)
97 \( 1 + 3.24e18T + 5.60e37T^{2} \)
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

−17.71908803027600728388308970187, −16.42090384686554858038011428180, −15.28076745459323787618053232565, −14.59452628036230690177004475246, −11.04805537288902009364569448549, −10.10811348840849128741716830583, −9.079579987360715001286263547447, −6.78658545712579754561770291646, −4.90294414953706549580834398284, −2.93784213039622852729748705199, 0.39161270295712337687007204622, 1.16526510462630504903205839916, 2.81591091515478260787988113159, 6.24364681540221598313492811432, 7.86411639376871843356230020265, 8.924073554095529609651799961039, 11.42724287683490453713071517586, 12.67577345725714192001078081916, 13.41324287370817643687037148213, 16.49675755667284244679655414099

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