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Properties

Label 2-2e3-8.3-c12-0-9
Degree $2$
Conductor $8$
Sign $-0.201 + 0.979i$
Analytic cond. $7.31195$
Root an. cond. $2.70406$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (53.1 + 35.6i)2-s − 1.02e3·3-s + (1.55e3 + 3.79e3i)4-s − 2.12e4i·5-s + (−5.43e4 − 3.64e4i)6-s − 1.49e5i·7-s + (−5.27e4 + 2.56e5i)8-s + 5.13e5·9-s + (7.57e5 − 1.12e6i)10-s − 2.66e6·11-s + (−1.58e6 − 3.87e6i)12-s − 2.70e6i·13-s + (5.31e6 − 7.92e6i)14-s + 2.17e7i·15-s + (−1.19e7 + 1.17e7i)16-s + 8.65e6·17-s + ⋯
L(s)  = 1  + (0.830 + 0.557i)2-s − 1.40·3-s + (0.378 + 0.925i)4-s − 1.35i·5-s + (−1.16 − 0.781i)6-s − 1.26i·7-s + (−0.201 + 0.979i)8-s + 0.965·9-s + (0.757 − 1.12i)10-s − 1.50·11-s + (−0.531 − 1.29i)12-s − 0.560i·13-s + (0.706 − 1.05i)14-s + 1.90i·15-s + (−0.712 + 0.701i)16-s + 0.358·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(7.31195\)
Root analytic conductor: \(2.70406\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :6),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.560108 - 0.686739i\)
\(L(\frac12)\) \(\approx\) \(0.560108 - 0.686739i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-53.1 - 35.6i)T \)
good3 \( 1 + 1.02e3T + 5.31e5T^{2} \)
5 \( 1 + 2.12e4iT - 2.44e8T^{2} \)
7 \( 1 + 1.49e5iT - 1.38e10T^{2} \)
11 \( 1 + 2.66e6T + 3.13e12T^{2} \)
13 \( 1 + 2.70e6iT - 2.32e13T^{2} \)
17 \( 1 - 8.65e6T + 5.82e14T^{2} \)
19 \( 1 + 3.18e7T + 2.21e15T^{2} \)
23 \( 1 - 7.29e7iT - 2.19e16T^{2} \)
29 \( 1 + 7.54e8iT - 3.53e17T^{2} \)
31 \( 1 - 1.62e8iT - 7.87e17T^{2} \)
37 \( 1 + 2.33e9iT - 6.58e18T^{2} \)
41 \( 1 - 5.58e9T + 2.25e19T^{2} \)
43 \( 1 + 1.62e9T + 3.99e19T^{2} \)
47 \( 1 + 4.97e9iT - 1.16e20T^{2} \)
53 \( 1 - 3.93e10iT - 4.91e20T^{2} \)
59 \( 1 + 2.97e10T + 1.77e21T^{2} \)
61 \( 1 + 3.50e10iT - 2.65e21T^{2} \)
67 \( 1 - 1.14e11T + 8.18e21T^{2} \)
71 \( 1 + 9.69e10iT - 1.64e22T^{2} \)
73 \( 1 + 1.09e11T + 2.29e22T^{2} \)
79 \( 1 + 3.71e11iT - 5.90e22T^{2} \)
83 \( 1 - 7.22e10T + 1.06e23T^{2} \)
89 \( 1 - 6.12e11T + 2.46e23T^{2} \)
97 \( 1 + 7.75e11T + 6.93e23T^{2} \)
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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

−17.56631227192521988103311959977, −16.78226632858329819783003430960, −15.82365760402029108359865200032, −13.37408571708996496666275275845, −12.43450528493197705347948905557, −10.74667719762928713250988562525, −7.74624632230807061860690738421, −5.68848182162305454268147159134, −4.54476408260015526892293406301, −0.41144271372840403216473021353, 2.59763227495288490600146869405, 5.28236838001925575274718265820, 6.51067907453677788855868915565, 10.40889845416627236327278512965, 11.36155646245390177119777847827, 12.63589462113446218659905267382, 14.69558778455118194214055597719, 15.94482214800026257586318765458, 18.22124798194949566503234638193, 18.81461763075066599997908122843

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