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Properties

Label 2-2e3-8.5-c7-0-1
Degree $2$
Conductor $8$
Sign $-0.401 - 0.915i$
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  + (1.55 + 11.2i)2-s + 21.9i·3-s + (−123. + 34.7i)4-s + 184. i·5-s + (−245. + 34.0i)6-s + 1.05e3·7-s + (−581. − 1.32e3i)8-s + 1.70e3·9-s + (−2.07e3 + 286. i)10-s − 4.32e3i·11-s + (−763. − 2.70e3i)12-s + 1.12e4i·13-s + (1.63e3 + 1.17e4i)14-s − 4.05e3·15-s + (1.39e4 − 8.57e3i)16-s − 2.17e4·17-s + ⋯
L(s)  = 1  + (0.137 + 0.990i)2-s + 0.469i·3-s + (−0.962 + 0.271i)4-s + 0.661i·5-s + (−0.464 + 0.0643i)6-s + 1.15·7-s + (−0.401 − 0.915i)8-s + 0.779·9-s + (−0.655 + 0.0907i)10-s − 0.979i·11-s + (−0.127 − 0.451i)12-s + 1.42i·13-s + (0.159 + 1.14i)14-s − 0.310·15-s + (0.852 − 0.523i)16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.401 - 0.915i$
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: \((2,\ 8,\ (\ :7/2),\ -0.401 - 0.915i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.756868 + 1.15780i\)
\(L(\frac12)\) \(\approx\) \(0.756868 + 1.15780i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.55 - 11.2i)T \)
good3 \( 1 - 21.9iT - 2.18e3T^{2} \)
5 \( 1 - 184. iT - 7.81e4T^{2} \)
7 \( 1 - 1.05e3T + 8.23e5T^{2} \)
11 \( 1 + 4.32e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.12e4iT - 6.27e7T^{2} \)
17 \( 1 + 2.17e4T + 4.10e8T^{2} \)
19 \( 1 + 4.54e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.41e3T + 3.40e9T^{2} \)
29 \( 1 + 2.36e4iT - 1.72e10T^{2} \)
31 \( 1 - 7.29e4T + 2.75e10T^{2} \)
37 \( 1 + 4.83e5iT - 9.49e10T^{2} \)
41 \( 1 + 4.11e5T + 1.94e11T^{2} \)
43 \( 1 - 9.61e4iT - 2.71e11T^{2} \)
47 \( 1 + 1.56e5T + 5.06e11T^{2} \)
53 \( 1 - 6.86e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.79e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.36e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.08e6iT - 6.06e12T^{2} \)
71 \( 1 + 5.60e6T + 9.09e12T^{2} \)
73 \( 1 - 2.16e4T + 1.10e13T^{2} \)
79 \( 1 - 2.34e6T + 1.92e13T^{2} \)
83 \( 1 - 8.82e5iT - 2.71e13T^{2} \)
89 \( 1 + 1.34e6T + 4.42e13T^{2} \)
97 \( 1 - 7.32e6T + 8.07e13T^{2} \)
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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

−21.35788348197290988797078466633, −18.90111925122497020179373733699, −17.68831667197428430799884663147, −16.12947103564453657801218988788, −14.87973269024281657895970548552, −13.64819902178425156841798752973, −11.11857004830857833873560851977, −8.929977715563553279398151659820, −6.92155637682498439363514104231, −4.53756424877732083482689976982, 1.51085202710998069083916310670, 4.76379405802049778035755527966, 8.181337482655758868344948114718, 10.27073147815272286823832202448, 12.12670376542152872730644551742, 13.19165952608890143793749264417, 15.00166651897654578970492760280, 17.49883637291075274668444759869, 18.42607595205701862638152056618, 20.19050146276976464968159861280

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