Properties

Degree 2
Conductor 3
Sign $-0.818 - 0.574i$
Motivic weight 16
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 312. i·2-s + (−5.36e3 − 3.77e3i)3-s − 3.22e4·4-s + 2.76e5i·5-s + (−1.17e6 + 1.67e6i)6-s − 7.10e6·7-s − 1.04e7i·8-s + (1.46e7 + 4.04e7i)9-s + 8.65e7·10-s − 3.43e8i·11-s + (1.73e8 + 1.21e8i)12-s − 7.14e8·13-s + 2.22e9i·14-s + (1.04e9 − 1.48e9i)15-s − 5.36e9·16-s − 6.74e8i·17-s + ⋯
L(s)  = 1  − 1.22i·2-s + (−0.818 − 0.574i)3-s − 0.492·4-s + 0.708i·5-s + (−0.701 + 0.999i)6-s − 1.23·7-s − 0.620i·8-s + (0.339 + 0.940i)9-s + 0.865·10-s − 1.60i·11-s + (0.402 + 0.282i)12-s − 0.876·13-s + 1.50i·14-s + (0.407 − 0.579i)15-s − 1.24·16-s − 0.0967i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-0.818 - 0.574i$
motivic weight  =  \(16\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :8),\ -0.818 - 0.574i)$
$L(\frac{17}{2})$  $\approx$  $0.189391 + 0.599352i$
$L(\frac12)$  $\approx$  $0.189391 + 0.599352i$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (5.36e3 + 3.77e3i)T \)
good2 \( 1 + 312. iT - 6.55e4T^{2} \)
5 \( 1 - 2.76e5iT - 1.52e11T^{2} \)
7 \( 1 + 7.10e6T + 3.32e13T^{2} \)
11 \( 1 + 3.43e8iT - 4.59e16T^{2} \)
13 \( 1 + 7.14e8T + 6.65e17T^{2} \)
17 \( 1 + 6.74e8iT - 4.86e19T^{2} \)
19 \( 1 - 4.70e9T + 2.88e20T^{2} \)
23 \( 1 + 4.47e10iT - 6.13e21T^{2} \)
29 \( 1 + 3.75e11iT - 2.50e23T^{2} \)
31 \( 1 - 6.58e11T + 7.27e23T^{2} \)
37 \( 1 - 3.89e11T + 1.23e25T^{2} \)
41 \( 1 - 4.14e12iT - 6.37e25T^{2} \)
43 \( 1 + 1.68e13T + 1.36e26T^{2} \)
47 \( 1 + 9.08e12iT - 5.66e26T^{2} \)
53 \( 1 + 4.32e13iT - 3.87e27T^{2} \)
59 \( 1 - 2.07e14iT - 2.15e28T^{2} \)
61 \( 1 - 1.10e14T + 3.67e28T^{2} \)
67 \( 1 - 1.61e14T + 1.64e29T^{2} \)
71 \( 1 + 8.06e13iT - 4.16e29T^{2} \)
73 \( 1 + 8.85e14T + 6.50e29T^{2} \)
79 \( 1 - 2.35e15T + 2.30e30T^{2} \)
83 \( 1 + 3.93e15iT - 5.07e30T^{2} \)
89 \( 1 + 3.56e15iT - 1.54e31T^{2} \)
97 \( 1 + 1.51e15T + 6.14e31T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.77955848001828004173888235131, −19.44938708742468372066300814206, −18.65617881665796415201514059676, −16.37787049749053713916100802538, −13.26976432337153163029235510906, −11.71164296627906324522090579035, −10.27199668072309756602977930882, −6.53455730748199774289560544157, −2.90750237650572410083716922065, −0.42218955733838694584384415923, 4.99031573645298447056423255859, 6.84783688008767126190920179119, 9.641806859839399850193019481738, 12.42351768649475129769376715329, 15.20664748669160562869486037550, 16.38522057107798428245705012450, 17.47825548440015866452895385767, 20.26672307508967734396696774218, 22.43281748559220214795500563929, 23.56523187292387208434549521472

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