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Properties

Label 2-2e3-8.5-c15-0-4
Degree $2$
Conductor $8$
Sign $-0.799 - 0.601i$
Analytic cond. $11.4154$
Root an. cond. $3.37868$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−176. − 38.6i)2-s + 4.90e3i·3-s + (2.97e4 + 1.36e4i)4-s + 1.74e5i·5-s + (1.89e5 − 8.66e5i)6-s + 3.39e6·7-s + (−4.73e6 − 3.56e6i)8-s − 9.66e6·9-s + (6.73e6 − 3.08e7i)10-s + 8.19e7i·11-s + (−6.69e7 + 1.45e8i)12-s + 1.80e8i·13-s + (−6.01e8 − 1.31e8i)14-s − 8.54e8·15-s + (7.00e8 + 8.13e8i)16-s − 1.02e9·17-s + ⋯
L(s)  = 1  + (−0.976 − 0.213i)2-s + 1.29i·3-s + (0.908 + 0.416i)4-s + 0.997i·5-s + (0.276 − 1.26i)6-s + 1.55·7-s + (−0.799 − 0.601i)8-s − 0.673·9-s + (0.212 − 0.974i)10-s + 1.26i·11-s + (−0.539 + 1.17i)12-s + 0.796i·13-s + (−1.52 − 0.332i)14-s − 1.29·15-s + (0.652 + 0.757i)16-s − 0.606·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.799 - 0.601i$
Analytic conductor: \(11.4154\)
Root analytic conductor: \(3.37868\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :15/2),\ -0.799 - 0.601i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.378569 + 1.13280i\)
\(L(\frac12)\) \(\approx\) \(0.378569 + 1.13280i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (176. + 38.6i)T \)
good3 \( 1 - 4.90e3iT - 1.43e7T^{2} \)
5 \( 1 - 1.74e5iT - 3.05e10T^{2} \)
7 \( 1 - 3.39e6T + 4.74e12T^{2} \)
11 \( 1 - 8.19e7iT - 4.17e15T^{2} \)
13 \( 1 - 1.80e8iT - 5.11e16T^{2} \)
17 \( 1 + 1.02e9T + 2.86e18T^{2} \)
19 \( 1 + 5.87e9iT - 1.51e19T^{2} \)
23 \( 1 + 9.43e9T + 2.66e20T^{2} \)
29 \( 1 + 7.53e10iT - 8.62e21T^{2} \)
31 \( 1 + 1.68e11T + 2.34e22T^{2} \)
37 \( 1 - 3.21e11iT - 3.33e23T^{2} \)
41 \( 1 - 1.60e12T + 1.55e24T^{2} \)
43 \( 1 + 1.36e12iT - 3.17e24T^{2} \)
47 \( 1 - 1.39e12T + 1.20e25T^{2} \)
53 \( 1 - 2.31e12iT - 7.31e25T^{2} \)
59 \( 1 - 2.85e13iT - 3.65e26T^{2} \)
61 \( 1 + 2.15e13iT - 6.02e26T^{2} \)
67 \( 1 - 7.75e12iT - 2.46e27T^{2} \)
71 \( 1 + 9.37e13T + 5.87e27T^{2} \)
73 \( 1 - 1.79e14T + 8.90e27T^{2} \)
79 \( 1 + 1.28e14T + 2.91e28T^{2} \)
83 \( 1 + 2.32e14iT - 6.11e28T^{2} \)
89 \( 1 - 4.72e12T + 1.74e29T^{2} \)
97 \( 1 - 4.41e14T + 6.33e29T^{2} \)
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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

−18.27003069720433300230544719133, −17.30611630238055762332395652386, −15.53984716267482610156394349533, −14.69733786768649091681517392601, −11.46173556494956805028833015473, −10.57299105315770158196015339045, −9.165597711316500188230980983279, −7.23739542368139822287028734723, −4.41529633684930909387399227714, −2.17396228112553412325439054013, 0.76692246943419293797308263204, 1.73888589386845511023474912886, 5.69952068943312332445893032626, 7.77790886362972993132241597004, 8.541259153557803897633927056261, 11.08225771375398468517674638826, 12.51291026120831825198419623407, 14.31688443762719449093613743513, 16.36122173092610453796058877724, 17.68772180349662495042784655300

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