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Properties

Label 2-2e3-8.3-c18-0-12
Degree $2$
Conductor $8$
Sign $-0.187 + 0.982i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−426. − 283. i)2-s + 3.58e4·3-s + (1.01e5 + 2.41e5i)4-s − 3.47e6i·5-s + (−1.52e7 − 1.01e7i)6-s + 1.08e7i·7-s + (2.52e7 − 1.31e8i)8-s + 8.97e8·9-s + (−9.85e8 + 1.48e9i)10-s − 1.36e8·11-s + (3.63e9 + 8.66e9i)12-s − 2.80e9i·13-s + (3.06e9 − 4.61e9i)14-s − 1.24e11i·15-s + (−4.81e10 + 4.90e10i)16-s + 2.99e10·17-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + 1.82·3-s + (0.387 + 0.922i)4-s − 1.78i·5-s + (−1.51 − 1.00i)6-s + 0.268i·7-s + (0.187 − 0.982i)8-s + 2.31·9-s + (−0.985 + 1.48i)10-s − 0.0577·11-s + (0.705 + 1.67i)12-s − 0.264i·13-s + (0.148 − 0.223i)14-s − 3.24i·15-s + (−0.700 + 0.713i)16-s + 0.252·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.46787 - 1.77546i\)
\(L(\frac12)\) \(\approx\) \(1.46787 - 1.77546i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (426. + 283. i)T \)
good3 \( 1 - 3.58e4T + 3.87e8T^{2} \)
5 \( 1 + 3.47e6iT - 3.81e12T^{2} \)
7 \( 1 - 1.08e7iT - 1.62e15T^{2} \)
11 \( 1 + 1.36e8T + 5.55e18T^{2} \)
13 \( 1 + 2.80e9iT - 1.12e20T^{2} \)
17 \( 1 - 2.99e10T + 1.40e22T^{2} \)
19 \( 1 + 5.03e10T + 1.04e23T^{2} \)
23 \( 1 + 1.64e12iT - 3.24e24T^{2} \)
29 \( 1 + 5.13e12iT - 2.10e26T^{2} \)
31 \( 1 + 3.37e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.22e14iT - 1.68e28T^{2} \)
41 \( 1 - 4.29e14T + 1.07e29T^{2} \)
43 \( 1 + 5.15e14T + 2.52e29T^{2} \)
47 \( 1 + 4.52e14iT - 1.25e30T^{2} \)
53 \( 1 - 3.85e15iT - 1.08e31T^{2} \)
59 \( 1 + 8.14e15T + 7.50e31T^{2} \)
61 \( 1 - 1.63e16iT - 1.36e32T^{2} \)
67 \( 1 - 3.10e16T + 7.40e32T^{2} \)
71 \( 1 - 4.98e16iT - 2.10e33T^{2} \)
73 \( 1 - 7.59e16T + 3.46e33T^{2} \)
79 \( 1 - 1.64e17iT - 1.43e34T^{2} \)
83 \( 1 - 4.04e16T + 3.49e34T^{2} \)
89 \( 1 - 1.65e17T + 1.22e35T^{2} \)
97 \( 1 + 4.02e17T + 5.77e35T^{2} \)
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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

−16.76871543610088431560059925786, −15.44743569225430745668807169081, −13.39756347453598621967637870605, −12.41236797594711461832283310708, −9.714402832002147251394626469386, −8.737502426085318633155598810615, −7.931832480560463408180986641746, −4.15207669644697006687077822898, −2.39142417526309707701107632855, −1.01357114893822156654922109032, 1.99299565008192176021165344889, 3.31715199403751993854475191095, 6.85607831687584768646295819476, 7.82976441653459631169097576023, 9.453068807808396708977514888219, 10.67564539013000388668529536801, 13.94921558378956945870849661164, 14.62997003340023349513187198593, 15.67759023102578537119428339083, 18.03725552741056117661479019524

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