Properties

Label 2-2e3-8.3-c12-0-2
Degree $2$
Conductor $8$
Sign $0.998 - 0.0588i$
Analytic cond. $7.31195$
Root an. cond. $2.70406$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−33.0 − 54.7i)2-s − 875.·3-s + (−1.90e3 + 3.62e3i)4-s + 2.27e3i·5-s + (2.89e4 + 4.79e4i)6-s − 9.20e4i·7-s + (2.61e5 − 1.54e4i)8-s + 2.34e5·9-s + (1.24e5 − 7.51e4i)10-s + 2.60e6·11-s + (1.66e6 − 3.17e6i)12-s + 8.33e6i·13-s + (−5.04e6 + 3.04e6i)14-s − 1.98e6i·15-s + (−9.50e6 − 1.38e7i)16-s − 7.99e6·17-s + ⋯
L(s)  = 1  + (−0.516 − 0.856i)2-s − 1.20·3-s + (−0.465 + 0.884i)4-s + 0.145i·5-s + (0.620 + 1.02i)6-s − 0.782i·7-s + (0.998 − 0.0588i)8-s + 0.442·9-s + (0.124 − 0.0751i)10-s + 1.46·11-s + (0.559 − 1.06i)12-s + 1.72i·13-s + (−0.669 + 0.404i)14-s − 0.174i·15-s + (−0.566 − 0.824i)16-s − 0.331·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.998 - 0.0588i$
Analytic conductor: \(7.31195\)
Root analytic conductor: \(2.70406\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :6),\ 0.998 - 0.0588i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.719944 + 0.0211933i\)
\(L(\frac12)\) \(\approx\) \(0.719944 + 0.0211933i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (33.0 + 54.7i)T \)
good3 \( 1 + 875.T + 5.31e5T^{2} \)
5 \( 1 - 2.27e3iT - 2.44e8T^{2} \)
7 \( 1 + 9.20e4iT - 1.38e10T^{2} \)
11 \( 1 - 2.60e6T + 3.13e12T^{2} \)
13 \( 1 - 8.33e6iT - 2.32e13T^{2} \)
17 \( 1 + 7.99e6T + 5.82e14T^{2} \)
19 \( 1 - 1.09e7T + 2.21e15T^{2} \)
23 \( 1 - 2.00e8iT - 2.19e16T^{2} \)
29 \( 1 - 1.82e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.49e9iT - 7.87e17T^{2} \)
37 \( 1 - 3.12e9iT - 6.58e18T^{2} \)
41 \( 1 + 2.93e9T + 2.25e19T^{2} \)
43 \( 1 - 4.63e9T + 3.99e19T^{2} \)
47 \( 1 - 3.74e9iT - 1.16e20T^{2} \)
53 \( 1 - 3.11e10iT - 4.91e20T^{2} \)
59 \( 1 + 2.18e9T + 1.77e21T^{2} \)
61 \( 1 + 3.86e9iT - 2.65e21T^{2} \)
67 \( 1 + 1.29e10T + 8.18e21T^{2} \)
71 \( 1 - 9.21e10iT - 1.64e22T^{2} \)
73 \( 1 - 2.54e11T + 2.29e22T^{2} \)
79 \( 1 - 2.54e11iT - 5.90e22T^{2} \)
83 \( 1 + 1.71e11T + 1.06e23T^{2} \)
89 \( 1 - 5.29e11T + 2.46e23T^{2} \)
97 \( 1 + 4.13e11T + 6.93e23T^{2} \)
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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

−18.78847482954254899788074648103, −17.20756004746534580617839426543, −16.71152067374131616199300085353, −13.86585601316675408317925948871, −11.89552712552648955007335100011, −11.12128987552837251175900432505, −9.375029701520913367404682165740, −6.83050288762459876463410491087, −4.19509184994293579437310821136, −1.19824296260177998244584975543, 0.66756268367327806591977822754, 5.22572352398713312068158411587, 6.50632904210740071021581079855, 8.744323450144724265588637799187, 10.65377963413421379364422572559, 12.37661938284524445040455827854, 14.67203804483544407511392199564, 16.12568365415874052602308111910, 17.31113227045644608244262005837, 18.19357226392170608308064390852

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