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Properties

Label 2-2e3-8.3-c16-0-10
Degree $2$
Conductor $8$
Sign $-0.147 + 0.989i$
Analytic cond. $12.9859$
Root an. cond. $3.60360$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−227. + 116. i)2-s + 6.14e3·3-s + (3.81e4 − 5.32e4i)4-s − 3.51e5i·5-s + (−1.39e6 + 7.18e5i)6-s + 1.24e6i·7-s + (−2.46e6 + 1.65e7i)8-s − 5.26e6·9-s + (4.10e7 + 7.99e7i)10-s − 2.06e8·11-s + (2.34e8 − 3.27e8i)12-s − 1.33e9i·13-s + (−1.46e8 − 2.84e8i)14-s − 2.15e9i·15-s + (−1.37e9 − 4.06e9i)16-s + 4.19e9·17-s + ⋯
L(s)  = 1  + (−0.889 + 0.456i)2-s + 0.936·3-s + (0.582 − 0.812i)4-s − 0.898i·5-s + (−0.833 + 0.427i)6-s + 0.216i·7-s + (−0.147 + 0.989i)8-s − 0.122·9-s + (0.410 + 0.799i)10-s − 0.963·11-s + (0.545 − 0.761i)12-s − 1.63i·13-s + (−0.0989 − 0.192i)14-s − 0.842i·15-s + (−0.320 − 0.947i)16-s + 0.601·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.147 + 0.989i$
Analytic conductor: \(12.9859\)
Root analytic conductor: \(3.60360\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :8),\ -0.147 + 0.989i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.672993 - 0.780540i\)
\(L(\frac12)\) \(\approx\) \(0.672993 - 0.780540i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (227. - 116. i)T \)
good3 \( 1 - 6.14e3T + 4.30e7T^{2} \)
5 \( 1 + 3.51e5iT - 1.52e11T^{2} \)
7 \( 1 - 1.24e6iT - 3.32e13T^{2} \)
11 \( 1 + 2.06e8T + 4.59e16T^{2} \)
13 \( 1 + 1.33e9iT - 6.65e17T^{2} \)
17 \( 1 - 4.19e9T + 4.86e19T^{2} \)
19 \( 1 + 1.43e10T + 2.88e20T^{2} \)
23 \( 1 + 1.00e11iT - 6.13e21T^{2} \)
29 \( 1 + 1.67e11iT - 2.50e23T^{2} \)
31 \( 1 - 1.59e11iT - 7.27e23T^{2} \)
37 \( 1 + 6.26e12iT - 1.23e25T^{2} \)
41 \( 1 + 1.26e13T + 6.37e25T^{2} \)
43 \( 1 - 2.52e12T + 1.36e26T^{2} \)
47 \( 1 - 3.05e13iT - 5.66e26T^{2} \)
53 \( 1 - 4.55e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.99e14T + 2.15e28T^{2} \)
61 \( 1 + 1.04e14iT - 3.67e28T^{2} \)
67 \( 1 - 6.31e14T + 1.64e29T^{2} \)
71 \( 1 - 9.94e14iT - 4.16e29T^{2} \)
73 \( 1 + 8.30e14T + 6.50e29T^{2} \)
79 \( 1 + 1.59e15iT - 2.30e30T^{2} \)
83 \( 1 + 5.20e14T + 5.07e30T^{2} \)
89 \( 1 - 1.46e15T + 1.54e31T^{2} \)
97 \( 1 - 4.30e15T + 6.14e31T^{2} \)
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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

−17.28382518807341099719866499802, −15.83919014956057021196444955597, −14.63688132958323655396535494953, −12.79590654857310450204979774100, −10.42088316097114319816089348572, −8.757026065996712702549188295510, −7.934251251587239945893913710573, −5.47420122528946500978618598556, −2.54630383088831262092897073057, −0.49387535176397551883411799072, 2.06393102272117719185744713970, 3.38928036440650936417518843262, 7.02626389904876756043320111397, 8.488389253037864512790260297276, 10.01898050900310952955807575556, 11.50282410647346882206880975886, 13.61106247621706474036514898340, 15.11069840652171939460324084728, 16.83782691489631463826255925010, 18.53971007372522727592282224264

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