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Properties

Label 2-2e3-8.3-c16-0-9
Degree $2$
Conductor $8$
Sign $0.834 - 0.551i$
Analytic cond. $12.9859$
Root an. cond. $3.60360$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (251. − 49.5i)2-s + 6.18e3·3-s + (6.06e4 − 2.48e4i)4-s + 6.99e5i·5-s + (1.55e6 − 3.06e5i)6-s + 4.65e6i·7-s + (1.39e7 − 9.25e6i)8-s − 4.83e6·9-s + (3.46e7 + 1.75e8i)10-s − 2.50e7·11-s + (3.74e8 − 1.53e8i)12-s − 9.99e8i·13-s + (2.30e8 + 1.16e9i)14-s + 4.32e9i·15-s + (3.05e9 − 3.01e9i)16-s + 1.78e9·17-s + ⋯
L(s)  = 1  + (0.981 − 0.193i)2-s + 0.942·3-s + (0.925 − 0.379i)4-s + 1.79i·5-s + (0.924 − 0.182i)6-s + 0.807i·7-s + (0.834 − 0.551i)8-s − 0.112·9-s + (0.346 + 1.75i)10-s − 0.117·11-s + (0.871 − 0.357i)12-s − 1.22i·13-s + (0.156 + 0.792i)14-s + 1.68i·15-s + (0.711 − 0.702i)16-s + 0.255·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(12.9859\)
Root analytic conductor: \(3.60360\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :8),\ 0.834 - 0.551i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(4.09899 + 1.23259i\)
\(L(\frac12)\) \(\approx\) \(4.09899 + 1.23259i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-251. + 49.5i)T \)
good3 \( 1 - 6.18e3T + 4.30e7T^{2} \)
5 \( 1 - 6.99e5iT - 1.52e11T^{2} \)
7 \( 1 - 4.65e6iT - 3.32e13T^{2} \)
11 \( 1 + 2.50e7T + 4.59e16T^{2} \)
13 \( 1 + 9.99e8iT - 6.65e17T^{2} \)
17 \( 1 - 1.78e9T + 4.86e19T^{2} \)
19 \( 1 - 2.70e10T + 2.88e20T^{2} \)
23 \( 1 + 7.64e10iT - 6.13e21T^{2} \)
29 \( 1 + 7.60e11iT - 2.50e23T^{2} \)
31 \( 1 - 2.35e11iT - 7.27e23T^{2} \)
37 \( 1 - 2.42e12iT - 1.23e25T^{2} \)
41 \( 1 - 1.88e12T + 6.37e25T^{2} \)
43 \( 1 + 9.47e12T + 1.36e26T^{2} \)
47 \( 1 - 1.60e13iT - 5.66e26T^{2} \)
53 \( 1 + 1.92e13iT - 3.87e27T^{2} \)
59 \( 1 + 3.49e13T + 2.15e28T^{2} \)
61 \( 1 - 3.45e14iT - 3.67e28T^{2} \)
67 \( 1 + 6.70e14T + 1.64e29T^{2} \)
71 \( 1 + 2.20e14iT - 4.16e29T^{2} \)
73 \( 1 - 1.81e14T + 6.50e29T^{2} \)
79 \( 1 + 1.45e14iT - 2.30e30T^{2} \)
83 \( 1 + 3.18e14T + 5.07e30T^{2} \)
89 \( 1 - 2.96e15T + 1.54e31T^{2} \)
97 \( 1 - 7.93e15T + 6.14e31T^{2} \)
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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

−18.28163703898439036840369633135, −15.44374511520395554054162949019, −14.72605480560166979749373170160, −13.65679294614958205749014938505, −11.69073979950641259383159514075, −10.17411332759761517280185851002, −7.65315365588609604238827568951, −5.90335578246807053430893050014, −3.19574391616774588058806912271, −2.57608937500916108963815037448, 1.49589473164828932950445092735, 3.70085141822402114349484022585, 5.16378312574966180114029319313, 7.63258636311930819200942911215, 9.160375186629920135743373990082, 11.83146487523482919456110635058, 13.32145100615922252563439563666, 14.14486153564947037106007509466, 16.01795203355927350341407397376, 16.93925672465394465230439615685

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