Properties

Label 2-2e3-8.3-c16-0-3
Degree $2$
Conductor $8$
Sign $0.538 - 0.842i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−48.2 + 251. i)2-s − 8.64e3·3-s + (−6.08e4 − 2.42e4i)4-s − 4.74e5i·5-s + (4.17e5 − 2.17e6i)6-s + 5.25e6i·7-s + (9.03e6 − 1.41e7i)8-s + 3.17e7·9-s + (1.19e8 + 2.28e7i)10-s − 3.36e8·11-s + (5.26e8 + 2.09e8i)12-s + 2.31e8i·13-s + (−1.32e9 − 2.53e8i)14-s + 4.10e9i·15-s + (3.11e9 + 2.95e9i)16-s + 6.30e9·17-s + ⋯
L(s)  = 1  + (−0.188 + 0.982i)2-s − 1.31·3-s + (−0.928 − 0.370i)4-s − 1.21i·5-s + (0.248 − 1.29i)6-s + 0.912i·7-s + (0.538 − 0.842i)8-s + 0.737·9-s + (1.19 + 0.228i)10-s − 1.56·11-s + (1.22 + 0.488i)12-s + 0.283i·13-s + (−0.895 − 0.172i)14-s + 1.59i·15-s + (0.725 + 0.687i)16-s + 0.903·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.538 - 0.842i$
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: \((2,\ 8,\ (\ :8),\ 0.538 - 0.842i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.615854 + 0.337151i\)
\(L(\frac12)\) \(\approx\) \(0.615854 + 0.337151i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (48.2 - 251. i)T \)
good3 \( 1 + 8.64e3T + 4.30e7T^{2} \)
5 \( 1 + 4.74e5iT - 1.52e11T^{2} \)
7 \( 1 - 5.25e6iT - 3.32e13T^{2} \)
11 \( 1 + 3.36e8T + 4.59e16T^{2} \)
13 \( 1 - 2.31e8iT - 6.65e17T^{2} \)
17 \( 1 - 6.30e9T + 4.86e19T^{2} \)
19 \( 1 - 2.41e10T + 2.88e20T^{2} \)
23 \( 1 - 8.96e10iT - 6.13e21T^{2} \)
29 \( 1 + 7.07e11iT - 2.50e23T^{2} \)
31 \( 1 + 7.96e11iT - 7.27e23T^{2} \)
37 \( 1 - 3.93e12iT - 1.23e25T^{2} \)
41 \( 1 + 2.45e12T + 6.37e25T^{2} \)
43 \( 1 - 7.81e12T + 1.36e26T^{2} \)
47 \( 1 - 1.93e13iT - 5.66e26T^{2} \)
53 \( 1 + 1.24e14iT - 3.87e27T^{2} \)
59 \( 1 - 6.48e13T + 2.15e28T^{2} \)
61 \( 1 - 1.48e14iT - 3.67e28T^{2} \)
67 \( 1 + 6.22e13T + 1.64e29T^{2} \)
71 \( 1 - 3.09e14iT - 4.16e29T^{2} \)
73 \( 1 - 7.83e14T + 6.50e29T^{2} \)
79 \( 1 - 7.76e14iT - 2.30e30T^{2} \)
83 \( 1 - 2.22e15T + 5.07e30T^{2} \)
89 \( 1 - 3.96e15T + 1.54e31T^{2} \)
97 \( 1 + 1.69e15T + 6.14e31T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.59910107454387504255862112603, −16.42617638439429137667119953953, −15.61241201498630618794728237845, −13.24493389731459742362005638006, −11.88652225365601641791144399694, −9.701291407538543364127549186806, −7.950472487032171658697043840584, −5.67990586044149512281247171384, −5.09835997668898320626865404970, −0.74702477533264377842603447325, 0.65478845134785127410169239897, 3.07087374864854425702341901013, 5.24375226294893415712309670074, 7.39678721911621090100976740911, 10.44353040389918361759851466350, 10.76372746917848032472304618104, 12.39437714338259075643646767653, 14.08257743373025813390724609107, 16.40613191072195418516798421005, 17.92027338783316787538927860519

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