Properties

Label 2-2e3-8.3-c20-0-6
Degree $2$
Conductor $8$
Sign $-0.997 - 0.0643i$
Analytic cond. $20.2811$
Root an. cond. $4.50345$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (492. + 897. i)2-s + 5.32e4·3-s + (−5.62e5 + 8.84e5i)4-s + 1.65e7i·5-s + (2.62e7 + 4.78e7i)6-s − 1.30e8i·7-s + (−1.07e9 − 6.90e7i)8-s − 6.47e8·9-s + (−1.48e10 + 8.16e9i)10-s + 2.41e10·11-s + (−2.99e10 + 4.71e10i)12-s + 6.71e10i·13-s + (1.16e11 − 6.41e10i)14-s + 8.83e11i·15-s + (−4.66e11 − 9.95e11i)16-s − 2.87e12·17-s + ⋯
L(s)  = 1  + (0.481 + 0.876i)2-s + 0.902·3-s + (−0.536 + 0.843i)4-s + 1.69i·5-s + (0.434 + 0.790i)6-s − 0.460i·7-s + (−0.997 − 0.0643i)8-s − 0.185·9-s + (−1.48 + 0.816i)10-s + 0.932·11-s + (−0.484 + 0.761i)12-s + 0.487i·13-s + (0.404 − 0.221i)14-s + 1.53i·15-s + (−0.423 − 0.905i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0643i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.997 - 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.997 - 0.0643i$
Analytic conductor: \(20.2811\)
Root analytic conductor: \(4.50345\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :10),\ -0.997 - 0.0643i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.0827514 + 2.57114i\)
\(L(\frac12)\) \(\approx\) \(0.0827514 + 2.57114i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-492. - 897. i)T \)
good3 \( 1 - 5.32e4T + 3.48e9T^{2} \)
5 \( 1 - 1.65e7iT - 9.53e13T^{2} \)
7 \( 1 + 1.30e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.41e10T + 6.72e20T^{2} \)
13 \( 1 - 6.71e10iT - 1.90e22T^{2} \)
17 \( 1 + 2.87e12T + 4.06e24T^{2} \)
19 \( 1 - 7.21e12T + 3.75e25T^{2} \)
23 \( 1 - 2.96e12iT - 1.71e27T^{2} \)
29 \( 1 - 4.38e14iT - 1.76e29T^{2} \)
31 \( 1 + 9.28e14iT - 6.71e29T^{2} \)
37 \( 1 - 7.26e15iT - 2.31e31T^{2} \)
41 \( 1 + 1.47e15T + 1.80e32T^{2} \)
43 \( 1 - 3.77e16T + 4.67e32T^{2} \)
47 \( 1 - 5.25e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.49e17iT - 3.05e34T^{2} \)
59 \( 1 - 2.43e17T + 2.61e35T^{2} \)
61 \( 1 + 5.14e17iT - 5.08e35T^{2} \)
67 \( 1 - 2.15e18T + 3.32e36T^{2} \)
71 \( 1 + 1.26e18iT - 1.05e37T^{2} \)
73 \( 1 - 3.77e18T + 1.84e37T^{2} \)
79 \( 1 + 2.34e18iT - 8.96e37T^{2} \)
83 \( 1 + 1.71e19T + 2.40e38T^{2} \)
89 \( 1 - 6.55e18T + 9.72e38T^{2} \)
97 \( 1 - 1.29e20T + 5.43e39T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.32470820111269509945167933111, −15.44427932862536050570814693965, −14.35587626601478279493706107133, −13.76429515643184755280456525613, −11.36606109108083434302818476597, −9.237081627827857598711834704853, −7.48844472144312999135689819728, −6.42208885791461242068690364092, −3.88892074048458811735393619751, −2.71115050930887708228136505817, 0.70988937495203352185632857709, 2.18530884784956870021133266016, 3.95626970103672275846626566286, 5.44213968276812057561659141071, 8.638926624159304893064916285064, 9.328985495671441352682584889870, 11.70765677745035093240475593892, 12.92471638028541442339177776479, 14.07823332576092237718745157960, 15.66982034086066641766070211011

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