Properties

Label 2-2e3-8.3-c14-0-2
Degree $2$
Conductor $8$
Sign $0.00708 - 0.999i$
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  + (−111. + 63.7i)2-s − 2.04e3·3-s + (8.25e3 − 1.41e4i)4-s − 5.48e4i·5-s + (2.26e5 − 1.30e5i)6-s − 4.32e5i·7-s + (−1.48e4 + 2.09e6i)8-s − 6.02e5·9-s + (3.49e6 + 6.09e6i)10-s − 5.00e6·11-s + (−1.68e7 + 2.89e7i)12-s + 1.07e8i·13-s + (2.75e7 + 4.79e7i)14-s + 1.12e8i·15-s + (−1.32e8 − 2.33e8i)16-s + 5.41e8·17-s + ⋯
L(s)  = 1  + (−0.867 + 0.497i)2-s − 0.934·3-s + (0.504 − 0.863i)4-s − 0.702i·5-s + (0.810 − 0.465i)6-s − 0.524i·7-s + (−0.00708 + 0.999i)8-s − 0.125·9-s + (0.349 + 0.609i)10-s − 0.256·11-s + (−0.471 + 0.807i)12-s + 1.70i·13-s + (0.261 + 0.455i)14-s + 0.656i·15-s + (−0.491 − 0.870i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00708 - 0.999i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.00708 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.00708 - 0.999i$
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: \((2,\ 8,\ (\ :7),\ 0.00708 - 0.999i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.377322 + 0.374658i\)
\(L(\frac12)\) \(\approx\) \(0.377322 + 0.374658i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (111. - 63.7i)T \)
good3 \( 1 + 2.04e3T + 4.78e6T^{2} \)
5 \( 1 + 5.48e4iT - 6.10e9T^{2} \)
7 \( 1 + 4.32e5iT - 6.78e11T^{2} \)
11 \( 1 + 5.00e6T + 3.79e14T^{2} \)
13 \( 1 - 1.07e8iT - 3.93e15T^{2} \)
17 \( 1 - 5.41e8T + 1.68e17T^{2} \)
19 \( 1 + 5.65e8T + 7.99e17T^{2} \)
23 \( 1 + 2.23e8iT - 1.15e19T^{2} \)
29 \( 1 - 2.03e10iT - 2.97e20T^{2} \)
31 \( 1 - 4.80e10iT - 7.56e20T^{2} \)
37 \( 1 - 2.61e9iT - 9.01e21T^{2} \)
41 \( 1 - 1.87e11T + 3.79e22T^{2} \)
43 \( 1 + 4.54e11T + 7.38e22T^{2} \)
47 \( 1 + 6.78e11iT - 2.56e23T^{2} \)
53 \( 1 - 1.93e11iT - 1.37e24T^{2} \)
59 \( 1 + 6.14e10T + 6.19e24T^{2} \)
61 \( 1 - 4.19e12iT - 9.87e24T^{2} \)
67 \( 1 + 2.70e12T + 3.67e25T^{2} \)
71 \( 1 + 7.21e12iT - 8.27e25T^{2} \)
73 \( 1 - 3.25e12T + 1.22e26T^{2} \)
79 \( 1 - 2.62e13iT - 3.68e26T^{2} \)
83 \( 1 - 2.72e13T + 7.36e26T^{2} \)
89 \( 1 + 8.38e13T + 1.95e27T^{2} \)
97 \( 1 - 9.36e12T + 6.52e27T^{2} \)
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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

−18.29372509833452978446171638637, −16.75390621929823848571829315937, −16.49465433364610675334628632469, −14.30897526737291239219372931126, −12.00408901278943729642222454934, −10.53523410683534134465755575843, −8.773175216350046486058595375431, −6.80173669531408073436453349891, −5.11881193279979874664001644606, −1.18042020569037904092589161901, 0.43802666569849847551025770718, 2.86078910865813491664096188288, 5.93184428892404752692459687113, 7.945492043065418701465206573723, 10.12339790684816735182185509648, 11.26589643566920008189608105222, 12.60931586572672772353329787782, 15.21488430494203085785437535447, 16.85144614646004757888482766935, 17.94012363534600760671663105993

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