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Properties

Label 2-2e3-8.3-c10-0-8
Degree $2$
Conductor $8$
Sign $-0.228 + 0.973i$
Analytic cond. $5.08285$
Root an. cond. $2.25451$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (28.8 − 13.8i)2-s − 196.·3-s + (641. − 798. i)4-s − 5.38e3i·5-s + (−5.67e3 + 2.71e3i)6-s + 6.61e3i·7-s + (7.47e3 − 3.19e4i)8-s − 2.03e4·9-s + (−7.44e4 − 1.55e5i)10-s + 2.21e5·11-s + (−1.26e5 + 1.56e5i)12-s + 3.61e5i·13-s + (9.14e4 + 1.90e5i)14-s + 1.05e6i·15-s + (−2.25e5 − 1.02e6i)16-s + 7.76e5·17-s + ⋯
L(s)  = 1  + (0.901 − 0.432i)2-s − 0.809·3-s + (0.626 − 0.779i)4-s − 1.72i·5-s + (−0.729 + 0.349i)6-s + 0.393i·7-s + (0.228 − 0.973i)8-s − 0.344·9-s + (−0.744 − 1.55i)10-s + 1.37·11-s + (−0.507 + 0.630i)12-s + 0.974i·13-s + (0.170 + 0.354i)14-s + 1.39i·15-s + (−0.215 − 0.976i)16-s + 0.546·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.228 + 0.973i$
Analytic conductor: \(5.08285\)
Root analytic conductor: \(2.25451\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :5),\ -0.228 + 0.973i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.20762 - 1.52323i\)
\(L(\frac12)\) \(\approx\) \(1.20762 - 1.52323i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-28.8 + 13.8i)T \)
good3 \( 1 + 196.T + 5.90e4T^{2} \)
5 \( 1 + 5.38e3iT - 9.76e6T^{2} \)
7 \( 1 - 6.61e3iT - 2.82e8T^{2} \)
11 \( 1 - 2.21e5T + 2.59e10T^{2} \)
13 \( 1 - 3.61e5iT - 1.37e11T^{2} \)
17 \( 1 - 7.76e5T + 2.01e12T^{2} \)
19 \( 1 - 1.46e6T + 6.13e12T^{2} \)
23 \( 1 + 2.34e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.88e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.00e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.16e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.70e7T + 1.34e16T^{2} \)
43 \( 1 - 3.52e7T + 2.16e16T^{2} \)
47 \( 1 - 4.23e8iT - 5.25e16T^{2} \)
53 \( 1 + 9.16e6iT - 1.74e17T^{2} \)
59 \( 1 + 1.03e9T + 5.11e17T^{2} \)
61 \( 1 + 1.37e8iT - 7.13e17T^{2} \)
67 \( 1 + 7.94e8T + 1.82e18T^{2} \)
71 \( 1 - 2.04e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.88e9T + 4.29e18T^{2} \)
79 \( 1 + 1.04e9iT - 9.46e18T^{2} \)
83 \( 1 - 3.77e9T + 1.55e19T^{2} \)
89 \( 1 + 4.07e9T + 3.11e19T^{2} \)
97 \( 1 - 1.76e9T + 7.37e19T^{2} \)
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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

−19.42018154588921149618958348666, −17.09488010200734778783735113807, −16.13555455165419127713244546709, −14.08466437818703087934857438695, −12.33254132398882362400657116391, −11.67173056203627760558484140625, −9.221588757400835911720773657783, −5.96685955414978732359088055394, −4.52544855981285131820693375679, −1.14962116667345822782600200498, 3.33854134247623960022608252835, 5.91629274648150166878305522810, 7.20427641006092238996840351375, 10.74886799677693933921105097839, 11.86746219357131094485037951801, 14.02278637976655651086662161876, 15.00317075864064807774279670860, 16.80707156048583088380150451527, 17.98134864088245443465164783460, 19.93158000937555625262494662400

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