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Properties

Label 2-2e3-8.3-c18-0-2
Degree $2$
Conductor $8$
Sign $-0.953 - 0.299i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−299. + 415. i)2-s + 5.93e3·3-s + (−8.25e4 − 2.48e5i)4-s − 2.28e6i·5-s + (−1.77e6 + 2.46e6i)6-s + 3.15e7i·7-s + (1.28e8 + 4.02e7i)8-s − 3.52e8·9-s + (9.47e8 + 6.83e8i)10-s + 2.35e9·11-s + (−4.90e8 − 1.47e9i)12-s + 1.20e10i·13-s + (−1.31e10 − 9.46e9i)14-s − 1.35e10i·15-s + (−5.50e10 + 4.10e10i)16-s − 1.97e11·17-s + ⋯
L(s)  = 1  + (−0.585 + 0.810i)2-s + 0.301·3-s + (−0.315 − 0.949i)4-s − 1.16i·5-s + (−0.176 + 0.244i)6-s + 0.782i·7-s + (0.953 + 0.299i)8-s − 0.908·9-s + (0.947 + 0.683i)10-s + 0.998·11-s + (−0.0950 − 0.286i)12-s + 1.13i·13-s + (−0.634 − 0.458i)14-s − 0.352i·15-s + (−0.801 + 0.598i)16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.953 - 0.299i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ -0.953 - 0.299i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.0855194 + 0.557188i\)
\(L(\frac12)\) \(\approx\) \(0.0855194 + 0.557188i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (299. - 415. i)T \)
good3 \( 1 - 5.93e3T + 3.87e8T^{2} \)
5 \( 1 + 2.28e6iT - 3.81e12T^{2} \)
7 \( 1 - 3.15e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.35e9T + 5.55e18T^{2} \)
13 \( 1 - 1.20e10iT - 1.12e20T^{2} \)
17 \( 1 + 1.97e11T + 1.40e22T^{2} \)
19 \( 1 + 1.48e11T + 1.04e23T^{2} \)
23 \( 1 - 1.21e12iT - 3.24e24T^{2} \)
29 \( 1 + 1.97e12iT - 2.10e26T^{2} \)
31 \( 1 - 2.27e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.66e14iT - 1.68e28T^{2} \)
41 \( 1 + 2.66e14T + 1.07e29T^{2} \)
43 \( 1 + 7.20e14T + 2.52e29T^{2} \)
47 \( 1 - 1.62e15iT - 1.25e30T^{2} \)
53 \( 1 - 2.17e15iT - 1.08e31T^{2} \)
59 \( 1 - 1.10e16T + 7.50e31T^{2} \)
61 \( 1 + 6.26e15iT - 1.36e32T^{2} \)
67 \( 1 + 4.55e16T + 7.40e32T^{2} \)
71 \( 1 + 5.40e16iT - 2.10e33T^{2} \)
73 \( 1 - 4.71e14T + 3.46e33T^{2} \)
79 \( 1 + 1.72e17iT - 1.43e34T^{2} \)
83 \( 1 + 8.17e16T + 3.49e34T^{2} \)
89 \( 1 - 4.18e17T + 1.22e35T^{2} \)
97 \( 1 + 5.20e17T + 5.77e35T^{2} \)
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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

−17.50192342140790337167486923256, −16.50689774136592223664900278339, −15.09681091832816807473635431924, −13.65369739025046581402907972963, −11.65096474313561704199596944798, −9.101383884462017101664631829406, −8.660185979669794930667128608018, −6.39158394613998981138378814671, −4.72014807740664297866971401199, −1.67770585309381920971517830821, 0.25080766117904725723032557294, 2.40721883812897041581624171666, 3.75987322373215949087351032185, 6.89126227282087735213999379511, 8.601249231017534116839385570492, 10.38367203682474591506143102799, 11.38981248683654252681420650068, 13.39764869412490313351905711893, 14.79777870184772321830758278526, 17.02128297819675251374726115574

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