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Properties

Label 2-2e3-8.5-c7-0-2
Degree $2$
Conductor $8$
Sign $0.874 - 0.484i$
Analytic cond. $2.49908$
Root an. cond. $1.58084$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.1 − 1.89i)2-s + 76.9i·3-s + (120. − 42.3i)4-s − 338. i·5-s + (146. + 858. i)6-s − 438.·7-s + (1.26e3 − 701. i)8-s − 3.73e3·9-s + (−642. − 3.77e3i)10-s − 1.96e3i·11-s + (3.25e3 + 9.29e3i)12-s + 2.21e3i·13-s + (−4.89e3 + 833. i)14-s + 2.60e4·15-s + (1.27e4 − 1.02e4i)16-s − 1.21e4·17-s + ⋯
L(s)  = 1  + (0.985 − 0.167i)2-s + 1.64i·3-s + (0.943 − 0.330i)4-s − 1.21i·5-s + (0.276 + 1.62i)6-s − 0.483·7-s + (0.874 − 0.484i)8-s − 1.70·9-s + (−0.203 − 1.19i)10-s − 0.445i·11-s + (0.544 + 1.55i)12-s + 0.279i·13-s + (−0.476 + 0.0811i)14-s + 1.99·15-s + (0.781 − 0.624i)16-s − 0.598·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(2.49908\)
Root analytic conductor: \(1.58084\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :7/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.03226 + 0.525088i\)
\(L(\frac12)\) \(\approx\) \(2.03226 + 0.525088i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.1 + 1.89i)T \)
good3 \( 1 - 76.9iT - 2.18e3T^{2} \)
5 \( 1 + 338. iT - 7.81e4T^{2} \)
7 \( 1 + 438.T + 8.23e5T^{2} \)
11 \( 1 + 1.96e3iT - 1.94e7T^{2} \)
13 \( 1 - 2.21e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.21e4T + 4.10e8T^{2} \)
19 \( 1 - 3.28e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.96e4T + 3.40e9T^{2} \)
29 \( 1 - 1.60e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.29e5T + 2.75e10T^{2} \)
37 \( 1 + 4.96e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.99e5T + 1.94e11T^{2} \)
43 \( 1 - 8.83e4iT - 2.71e11T^{2} \)
47 \( 1 - 8.20e5T + 5.06e11T^{2} \)
53 \( 1 + 1.53e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.82e6iT - 2.48e12T^{2} \)
61 \( 1 + 4.84e5iT - 3.14e12T^{2} \)
67 \( 1 + 7.98e4iT - 6.06e12T^{2} \)
71 \( 1 - 1.27e6T + 9.09e12T^{2} \)
73 \( 1 - 3.70e6T + 1.10e13T^{2} \)
79 \( 1 + 2.55e6T + 1.92e13T^{2} \)
83 \( 1 - 1.53e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.99e6T + 4.42e13T^{2} \)
97 \( 1 + 2.89e4T + 8.07e13T^{2} \)
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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

−20.80171470372449410001452017957, −19.81189732674982952448719114743, −16.52067291856575154371148826206, −16.06763636558915706067489396066, −14.50289024648326738785047472061, −12.68923739745156877346372992664, −10.87022029681820199795643416411, −9.200110809473735845947755289768, −5.39503189426215931113478803453, −3.90542560971273365534469377252, 2.59142600077458717024068467088, 6.40173411349288401009554966219, 7.38335003345176690415669664645, 11.20552536188170874799064326826, 12.72636488177693187857388383102, 13.79886666147489102896624994908, 15.21566584894568182624154320926, 17.48167897903956534140062200997, 18.83126652019151202879233135941, 20.00038387066108865761859832689

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