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Properties

Label 2-2e3-8.3-c14-0-11
Degree $2$
Conductor $8$
Sign $-0.980 - 0.196i$
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  + (−56.5 − 114. i)2-s + 1.52e3·3-s + (−9.98e3 + 1.29e4i)4-s − 7.16e4i·5-s + (−8.62e4 − 1.75e5i)6-s − 6.47e5i·7-s + (2.05e6 + 4.11e5i)8-s − 2.45e6·9-s + (−8.22e6 + 4.05e6i)10-s − 2.62e7·11-s + (−1.52e7 + 1.98e7i)12-s + 4.43e7i·13-s + (−7.43e7 + 3.66e7i)14-s − 1.09e8i·15-s + (−6.90e7 − 2.59e8i)16-s − 4.73e8·17-s + ⋯
L(s)  = 1  + (−0.441 − 0.897i)2-s + 0.697·3-s + (−0.609 + 0.792i)4-s − 0.916i·5-s + (−0.308 − 0.625i)6-s − 0.786i·7-s + (0.980 + 0.196i)8-s − 0.513·9-s + (−0.822 + 0.405i)10-s − 1.34·11-s + (−0.425 + 0.553i)12-s + 0.707i·13-s + (−0.705 + 0.347i)14-s − 0.639i·15-s + (−0.257 − 0.966i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.980 - 0.196i$
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: \((2,\ 8,\ (\ :7),\ -0.980 - 0.196i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.0820534 + 0.828026i\)
\(L(\frac12)\) \(\approx\) \(0.0820534 + 0.828026i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (56.5 + 114. i)T \)
good3 \( 1 - 1.52e3T + 4.78e6T^{2} \)
5 \( 1 + 7.16e4iT - 6.10e9T^{2} \)
7 \( 1 + 6.47e5iT - 6.78e11T^{2} \)
11 \( 1 + 2.62e7T + 3.79e14T^{2} \)
13 \( 1 - 4.43e7iT - 3.93e15T^{2} \)
17 \( 1 + 4.73e8T + 1.68e17T^{2} \)
19 \( 1 - 2.43e8T + 7.99e17T^{2} \)
23 \( 1 + 4.19e9iT - 1.15e19T^{2} \)
29 \( 1 + 2.44e10iT - 2.97e20T^{2} \)
31 \( 1 - 1.22e10iT - 7.56e20T^{2} \)
37 \( 1 + 6.12e10iT - 9.01e21T^{2} \)
41 \( 1 - 3.09e11T + 3.79e22T^{2} \)
43 \( 1 - 2.18e11T + 7.38e22T^{2} \)
47 \( 1 - 8.36e11iT - 2.56e23T^{2} \)
53 \( 1 + 2.10e12iT - 1.37e24T^{2} \)
59 \( 1 + 2.27e12T + 6.19e24T^{2} \)
61 \( 1 + 2.99e12iT - 9.87e24T^{2} \)
67 \( 1 + 8.46e11T + 3.67e25T^{2} \)
71 \( 1 + 1.11e13iT - 8.27e25T^{2} \)
73 \( 1 + 9.55e12T + 1.22e26T^{2} \)
79 \( 1 - 2.17e13iT - 3.68e26T^{2} \)
83 \( 1 - 1.17e13T + 7.36e26T^{2} \)
89 \( 1 - 5.85e13T + 1.95e27T^{2} \)
97 \( 1 + 1.22e14T + 6.52e27T^{2} \)
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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

−17.62389397310493191860091767481, −16.26625806412726092870759703942, −13.90442469800125705154219848476, −12.79802870991901624512516660134, −10.92300982972682536801542903490, −9.190423504490963367947880921351, −7.967100916108783109535018108853, −4.43417490613463230657936631436, −2.43070822665345682155845422073, −0.39965665839534092759246854712, 2.69470287687768787472521144226, 5.61966990855481100768249116994, 7.54781488472035632677284449707, 8.947255249849256554780886209096, 10.72365126898244089877586479842, 13.44159752727582413451023912215, 14.86130914522788687400046654564, 15.67647107693121027584246516484, 17.77579317660915997418120260928, 18.68434647099792238994780380071

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