Properties

Label 2-2e3-8.5-c15-0-9
Degree $2$
Conductor $8$
Sign $0.965 + 0.259i$
Analytic cond. $11.4154$
Root an. cond. $3.37868$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−103. + 148. i)2-s + 6.35e3i·3-s + (−1.11e4 − 3.07e4i)4-s − 2.67e5i·5-s + (−9.42e5 − 6.60e5i)6-s + 1.20e5·7-s + (5.72e6 + 1.53e6i)8-s − 2.60e7·9-s + (3.96e7 + 2.77e7i)10-s − 9.60e7i·11-s + (1.95e8 − 7.11e7i)12-s + 1.64e8i·13-s + (−1.25e7 + 1.78e7i)14-s + 1.69e9·15-s + (−8.23e8 + 6.89e8i)16-s − 1.25e8·17-s + ⋯
L(s)  = 1  + (−0.573 + 0.818i)2-s + 1.67i·3-s + (−0.341 − 0.939i)4-s − 1.52i·5-s + (−1.37 − 0.963i)6-s + 0.0553·7-s + (0.965 + 0.259i)8-s − 1.81·9-s + (1.25 + 0.877i)10-s − 1.48i·11-s + (1.57 − 0.573i)12-s + 0.724i·13-s + (−0.0317 + 0.0453i)14-s + 2.56·15-s + (−0.766 + 0.641i)16-s − 0.0741·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(11.4154\)
Root analytic conductor: \(3.37868\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :15/2),\ 0.965 + 0.259i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.836711 - 0.110509i\)
\(L(\frac12)\) \(\approx\) \(0.836711 - 0.110509i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (103. - 148. i)T \)
good3 \( 1 - 6.35e3iT - 1.43e7T^{2} \)
5 \( 1 + 2.67e5iT - 3.05e10T^{2} \)
7 \( 1 - 1.20e5T + 4.74e12T^{2} \)
11 \( 1 + 9.60e7iT - 4.17e15T^{2} \)
13 \( 1 - 1.64e8iT - 5.11e16T^{2} \)
17 \( 1 + 1.25e8T + 2.86e18T^{2} \)
19 \( 1 + 4.61e9iT - 1.51e19T^{2} \)
23 \( 1 - 6.78e9T + 2.66e20T^{2} \)
29 \( 1 + 7.66e10iT - 8.62e21T^{2} \)
31 \( 1 - 2.28e10T + 2.34e22T^{2} \)
37 \( 1 + 8.61e11iT - 3.33e23T^{2} \)
41 \( 1 + 3.52e11T + 1.55e24T^{2} \)
43 \( 1 + 1.68e12iT - 3.17e24T^{2} \)
47 \( 1 - 3.84e12T + 1.20e25T^{2} \)
53 \( 1 - 9.03e12iT - 7.31e25T^{2} \)
59 \( 1 + 1.13e13iT - 3.65e26T^{2} \)
61 \( 1 - 3.96e12iT - 6.02e26T^{2} \)
67 \( 1 + 4.34e13iT - 2.46e27T^{2} \)
71 \( 1 + 3.11e13T + 5.87e27T^{2} \)
73 \( 1 + 8.34e13T + 8.90e27T^{2} \)
79 \( 1 - 7.31e12T + 2.91e28T^{2} \)
83 \( 1 - 3.96e14iT - 6.11e28T^{2} \)
89 \( 1 - 9.54e13T + 1.74e29T^{2} \)
97 \( 1 + 1.16e15T + 6.33e29T^{2} \)
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

−17.11733054776379758598113540111, −16.37254612577655248054197612929, −15.54150043312060687172694209284, −13.80538026180202864571571238299, −11.06298674511715819013245881185, −9.315701504003479694916314838923, −8.642026220360304845824173608931, −5.53768025987212631504585161681, −4.36342045316022545357964009653, −0.47290295042212204902833781802, 1.60125323849680602666766926501, 2.88186768047336702522119847633, 6.82748399361251572503223034411, 7.82996083830182941401044733681, 10.28575523138334932156148065971, 11.84037190527099560986349523863, 13.01979014334411385908027849538, 14.58985813021447369291107774581, 17.53147597009598259482843883004, 18.26104436025198146674509975181

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