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Properties

Label 2-2e3-8.5-c11-0-5
Degree $2$
Conductor $8$
Sign $0.985 + 0.170i$
Analytic cond. $6.14674$
Root an. cond. $2.47926$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−24.8 + 37.8i)2-s − 102. i·3-s + (−814. − 1.87e3i)4-s − 2.31e3i·5-s + (3.86e3 + 2.54e3i)6-s + 3.81e4·7-s + (9.13e4 + 1.58e4i)8-s + 1.66e5·9-s + (8.77e4 + 5.76e4i)10-s − 3.22e5i·11-s + (−1.92e5 + 8.33e4i)12-s − 1.18e6i·13-s + (−9.47e5 + 1.44e6i)14-s − 2.37e5·15-s + (−2.86e6 + 3.06e6i)16-s + 1.63e6·17-s + ⋯
L(s)  = 1  + (−0.548 + 0.836i)2-s − 0.243i·3-s + (−0.397 − 0.917i)4-s − 0.331i·5-s + (0.203 + 0.133i)6-s + 0.858·7-s + (0.985 + 0.170i)8-s + 0.940·9-s + (0.277 + 0.182i)10-s − 0.604i·11-s + (−0.222 + 0.0966i)12-s − 0.888i·13-s + (−0.470 + 0.717i)14-s − 0.0806·15-s + (−0.683 + 0.729i)16-s + 0.278·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(6.14674\)
Root analytic conductor: \(2.47926\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :11/2),\ 0.985 + 0.170i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.28602 - 0.110691i\)
\(L(\frac12)\) \(\approx\) \(1.28602 - 0.110691i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (24.8 - 37.8i)T \)
good3 \( 1 + 102. iT - 1.77e5T^{2} \)
5 \( 1 + 2.31e3iT - 4.88e7T^{2} \)
7 \( 1 - 3.81e4T + 1.97e9T^{2} \)
11 \( 1 + 3.22e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.18e6iT - 1.79e12T^{2} \)
17 \( 1 - 1.63e6T + 3.42e13T^{2} \)
19 \( 1 + 1.58e7iT - 1.16e14T^{2} \)
23 \( 1 - 7.84e6T + 9.52e14T^{2} \)
29 \( 1 - 1.88e8iT - 1.22e16T^{2} \)
31 \( 1 + 2.59e8T + 2.54e16T^{2} \)
37 \( 1 - 5.80e8iT - 1.77e17T^{2} \)
41 \( 1 - 9.39e8T + 5.50e17T^{2} \)
43 \( 1 + 1.40e9iT - 9.29e17T^{2} \)
47 \( 1 - 3.57e8T + 2.47e18T^{2} \)
53 \( 1 + 1.56e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.83e9iT - 3.01e19T^{2} \)
61 \( 1 + 2.32e9iT - 4.35e19T^{2} \)
67 \( 1 - 1.50e10iT - 1.22e20T^{2} \)
71 \( 1 - 3.46e9T + 2.31e20T^{2} \)
73 \( 1 + 1.94e10T + 3.13e20T^{2} \)
79 \( 1 + 1.13e10T + 7.47e20T^{2} \)
83 \( 1 + 2.66e10iT - 1.28e21T^{2} \)
89 \( 1 - 4.93e10T + 2.77e21T^{2} \)
97 \( 1 + 1.79e10T + 7.15e21T^{2} \)
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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

−18.65545930796988717900485956555, −17.61962515752354987096742982379, −16.14773497031583124299769784085, −14.77664192906811962549524043595, −13.11627683788458329519768274465, −10.74590027180687984746270958675, −8.791939602930975843188376454153, −7.24616471005248635799874139691, −5.09078107257912521288729077565, −1.05586510734133990453650768935, 1.73096131922362252377117402114, 4.21981240726795754984814305572, 7.60012686410565149453435941826, 9.573866878370966534774279339144, 11.00098644860640566478866126089, 12.54578185014328588406802286915, 14.48043776569041795204744517222, 16.46552970026788476804355339158, 18.00257464720884748157974121996, 19.00670413096541726581731555266

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