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Properties

Label 2-2e3-8.5-c3-0-0
Degree $2$
Conductor $8$
Sign $0.883 - 0.467i$
Analytic cond. $0.472015$
Root an. cond. $0.687033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 2.64i)2-s − 5.29i·3-s + (−6.00 − 5.29i)4-s + 10.5i·5-s + (14.0 + 5.29i)6-s − 8·7-s + (20.0 − 10.5i)8-s − 1.00·9-s + (−28.0 − 10.5i)10-s + 15.8i·11-s + (−28.0 + 31.7i)12-s − 52.9i·13-s + (8 − 21.1i)14-s + 56.0·15-s + (8.00 + 63.4i)16-s − 14·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s − 1.01i·3-s + (−0.750 − 0.661i)4-s + 0.946i·5-s + (0.952 + 0.360i)6-s − 0.431·7-s + (0.883 − 0.467i)8-s − 0.0370·9-s + (−0.885 − 0.334i)10-s + 0.435i·11-s + (−0.673 + 0.763i)12-s − 1.12i·13-s + (0.152 − 0.404i)14-s + 0.963·15-s + (0.125 + 0.992i)16-s − 0.199·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.883 - 0.467i$
Analytic conductor: \(0.472015\)
Root analytic conductor: \(0.687033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :3/2),\ 0.883 - 0.467i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.644021 + 0.159889i\)
\(L(\frac12)\) \(\approx\) \(0.644021 + 0.159889i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 2.64i)T \)
good3 \( 1 + 5.29iT - 27T^{2} \)
5 \( 1 - 10.5iT - 125T^{2} \)
7 \( 1 + 8T + 343T^{2} \)
11 \( 1 - 15.8iT - 1.33e3T^{2} \)
13 \( 1 + 52.9iT - 2.19e3T^{2} \)
17 \( 1 + 14T + 4.91e3T^{2} \)
19 \( 1 - 37.0iT - 6.85e3T^{2} \)
23 \( 1 + 152T + 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 - 224T + 2.97e4T^{2} \)
37 \( 1 + 243. iT - 5.06e4T^{2} \)
41 \( 1 + 70T + 6.89e4T^{2} \)
43 \( 1 - 439. iT - 7.95e4T^{2} \)
47 \( 1 - 336T + 1.03e5T^{2} \)
53 \( 1 + 31.7iT - 1.48e5T^{2} \)
59 \( 1 + 534. iT - 2.05e5T^{2} \)
61 \( 1 + 95.2iT - 2.26e5T^{2} \)
67 \( 1 + 174. iT - 3.00e5T^{2} \)
71 \( 1 + 72T + 3.57e5T^{2} \)
73 \( 1 + 294T + 3.89e5T^{2} \)
79 \( 1 + 464T + 4.93e5T^{2} \)
83 \( 1 - 545. iT - 5.71e5T^{2} \)
89 \( 1 - 266T + 7.04e5T^{2} \)
97 \( 1 - 994T + 9.12e5T^{2} \)
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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

−22.32328609141571665667068374535, −19.68782470900856199067021394123, −18.47465037908060946616208160159, −17.71952466704451273167168990255, −15.76870394334660738848907925024, −14.30533482400885281017161563303, −12.80093509323794710250940362899, −10.19856353228435028276780554201, −7.76460757569393509809552959748, −6.41965094825064143108164280420, 4.35379543989648781937842455545, 8.832485509020613984625872254045, 10.06406893318100641723738699310, 11.87562116271811096424835068025, 13.56118051764597305259579034934, 15.95486277328282696300256864193, 17.01365951247526908992298118503, 18.94139912399034712115163287588, 20.32217722439891987038930398912, 21.30376301435529161239728484600

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