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Properties

Label 2-2e3-8.5-c11-0-9
Degree $2$
Conductor $8$
Sign $-0.984 + 0.176i$
Analytic cond. $6.14674$
Root an. cond. $2.47926$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (20.2 − 40.4i)2-s − 712. i·3-s + (−1.22e3 − 1.64e3i)4-s + 7.12e3i·5-s + (−2.88e4 − 1.44e4i)6-s + 4.69e4·7-s + (−9.12e4 + 1.63e4i)8-s − 3.29e5·9-s + (2.88e5 + 1.44e5i)10-s − 6.77e5i·11-s + (−1.16e6 + 8.73e5i)12-s − 4.38e5i·13-s + (9.50e5 − 1.89e6i)14-s + 5.07e6·15-s + (−1.18e6 + 4.02e6i)16-s + 5.66e6·17-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s − 1.69i·3-s + (−0.598 − 0.800i)4-s + 1.01i·5-s + (−1.51 − 0.757i)6-s + 1.05·7-s + (−0.984 + 0.176i)8-s − 1.86·9-s + (0.911 + 0.456i)10-s − 1.26i·11-s + (−1.35 + 1.01i)12-s − 0.327i·13-s + (0.472 − 0.943i)14-s + 1.72·15-s + (−0.282 + 0.959i)16-s + 0.967·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(6.14674\)
Root analytic conductor: \(2.47926\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :11/2),\ -0.984 + 0.176i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.163999 - 1.83974i\)
\(L(\frac12)\) \(\approx\) \(0.163999 - 1.83974i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-20.2 + 40.4i)T \)
good3 \( 1 + 712. iT - 1.77e5T^{2} \)
5 \( 1 - 7.12e3iT - 4.88e7T^{2} \)
7 \( 1 - 4.69e4T + 1.97e9T^{2} \)
11 \( 1 + 6.77e5iT - 2.85e11T^{2} \)
13 \( 1 + 4.38e5iT - 1.79e12T^{2} \)
17 \( 1 - 5.66e6T + 3.42e13T^{2} \)
19 \( 1 + 2.20e6iT - 1.16e14T^{2} \)
23 \( 1 + 2.68e7T + 9.52e14T^{2} \)
29 \( 1 + 4.65e6iT - 1.22e16T^{2} \)
31 \( 1 - 2.65e8T + 2.54e16T^{2} \)
37 \( 1 + 2.71e8iT - 1.77e17T^{2} \)
41 \( 1 - 6.32e8T + 5.50e17T^{2} \)
43 \( 1 - 1.17e9iT - 9.29e17T^{2} \)
47 \( 1 - 3.27e8T + 2.47e18T^{2} \)
53 \( 1 - 3.75e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.31e9iT - 3.01e19T^{2} \)
61 \( 1 - 4.24e9iT - 4.35e19T^{2} \)
67 \( 1 + 7.41e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.51e10T + 2.31e20T^{2} \)
73 \( 1 - 9.14e9T + 3.13e20T^{2} \)
79 \( 1 - 4.97e10T + 7.47e20T^{2} \)
83 \( 1 + 1.15e10iT - 1.28e21T^{2} \)
89 \( 1 + 8.86e10T + 2.77e21T^{2} \)
97 \( 1 + 1.71e10T + 7.15e21T^{2} \)
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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

−18.61238740359672518519817586468, −17.78590965707267749329759264352, −14.47857729584869633302861823399, −13.68179933591812552374930864963, −12.04368053873886821201844353137, −10.93347263178154298684516557149, −8.030496329685941066506949008374, −6.03535104036681206152845830956, −2.74515053180284408355831802381, −1.04170393479732175972278422105, 4.28282134250115599272602894963, 5.15245460957330976067858322575, 8.284391238342362881158530024142, 9.756718274268024600407852576151, 12.06537350611785443026099349064, 14.29584133500898568029402825953, 15.37976811647996632895276009187, 16.53160616953674555021147272079, 17.52348615761909959683825704365, 20.61678879269524471603793953606

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