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Properties

Label 2-2e3-8.3-c14-0-6
Degree $2$
Conductor $8$
Sign $0.633 + 0.773i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (29.0 + 124. i)2-s − 3.87e3·3-s + (−1.47e4 + 7.23e3i)4-s + 1.09e5i·5-s + (−1.12e5 − 4.83e5i)6-s − 6.42e5i·7-s + (−1.32e6 − 1.62e6i)8-s + 1.02e7·9-s + (−1.36e7 + 3.18e6i)10-s + 3.00e6·11-s + (5.70e7 − 2.80e7i)12-s + 3.85e7i·13-s + (8.01e7 − 1.86e7i)14-s − 4.26e8i·15-s + (1.63e8 − 2.12e8i)16-s − 5.22e8·17-s + ⋯
L(s)  = 1  + (0.226 + 0.973i)2-s − 1.77·3-s + (−0.897 + 0.441i)4-s + 1.40i·5-s + (−0.402 − 1.72i)6-s − 0.780i·7-s + (−0.633 − 0.773i)8-s + 2.14·9-s + (−1.36 + 0.318i)10-s + 0.153·11-s + (1.59 − 0.783i)12-s + 0.613i·13-s + (0.760 − 0.176i)14-s − 2.49i·15-s + (0.610 − 0.792i)16-s − 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.633 + 0.773i$
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: \((2,\ 8,\ (\ :7),\ 0.633 + 0.773i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.134707 - 0.0638136i\)
\(L(\frac12)\) \(\approx\) \(0.134707 - 0.0638136i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-29.0 - 124. i)T \)
good3 \( 1 + 3.87e3T + 4.78e6T^{2} \)
5 \( 1 - 1.09e5iT - 6.10e9T^{2} \)
7 \( 1 + 6.42e5iT - 6.78e11T^{2} \)
11 \( 1 - 3.00e6T + 3.79e14T^{2} \)
13 \( 1 - 3.85e7iT - 3.93e15T^{2} \)
17 \( 1 + 5.22e8T + 1.68e17T^{2} \)
19 \( 1 - 1.50e8T + 7.99e17T^{2} \)
23 \( 1 + 2.13e9iT - 1.15e19T^{2} \)
29 \( 1 + 3.24e10iT - 2.97e20T^{2} \)
31 \( 1 - 7.00e9iT - 7.56e20T^{2} \)
37 \( 1 - 8.58e10iT - 9.01e21T^{2} \)
41 \( 1 - 1.24e9T + 3.79e22T^{2} \)
43 \( 1 + 1.11e11T + 7.38e22T^{2} \)
47 \( 1 + 7.90e11iT - 2.56e23T^{2} \)
53 \( 1 + 6.09e11iT - 1.37e24T^{2} \)
59 \( 1 + 3.76e12T + 6.19e24T^{2} \)
61 \( 1 - 2.10e12iT - 9.87e24T^{2} \)
67 \( 1 + 3.54e12T + 3.67e25T^{2} \)
71 \( 1 + 1.52e12iT - 8.27e25T^{2} \)
73 \( 1 + 6.85e12T + 1.22e26T^{2} \)
79 \( 1 + 1.20e13iT - 3.68e26T^{2} \)
83 \( 1 + 3.52e13T + 7.36e26T^{2} \)
89 \( 1 - 1.62e13T + 1.95e27T^{2} \)
97 \( 1 - 2.55e13T + 6.52e27T^{2} \)
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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

−17.67868369107844216400386617825, −16.70280987709897940511690515180, −15.29264626575556119286201565402, −13.58228365847615828450244702499, −11.61629085394513534912161738098, −10.28164859504583260120497355865, −7.05366684086105345588123204584, −6.28805838328429022930134705827, −4.35417033741898978018038070960, −0.092419711721898985049194595582, 1.29245010746599129852731045784, 4.68840070991826834492406878565, 5.68438711647238425082369636201, 9.127984537317728521056605028248, 10.95012476233744934665478877326, 12.19093492539489946401558817096, 12.93046735897328805942873885052, 15.79970158213203884978150776339, 17.25243618769597567249079361029, 18.22837149729754616269930183642

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