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Properties

Label 2-2e3-8.3-c18-0-5
Degree $2$
Conductor $8$
Sign $0.959 + 0.283i$
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  + (297. − 416. i)2-s − 2.59e4·3-s + (−8.54e4 − 2.47e5i)4-s + 1.90e6i·5-s + (−7.70e6 + 1.08e7i)6-s + 9.33e6i·7-s + (−1.28e8 − 3.80e7i)8-s + 2.84e8·9-s + (7.93e8 + 5.65e8i)10-s + 2.94e9·11-s + (2.21e9 + 6.42e9i)12-s − 7.67e9i·13-s + (3.89e9 + 2.77e9i)14-s − 4.93e10i·15-s + (−5.41e10 + 4.23e10i)16-s − 1.40e10·17-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s − 1.31·3-s + (−0.326 − 0.945i)4-s + 0.974i·5-s + (−0.764 + 1.07i)6-s + 0.231i·7-s + (−0.959 − 0.283i)8-s + 0.734·9-s + (0.793 + 0.565i)10-s + 1.24·11-s + (0.429 + 1.24i)12-s − 0.723i·13-s + (0.188 + 0.134i)14-s − 1.28i·15-s + (−0.787 + 0.616i)16-s − 0.118·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.959 + 0.283i$
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: \((2,\ 8,\ (\ :9),\ 0.959 + 0.283i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.45521 - 0.210395i\)
\(L(\frac12)\) \(\approx\) \(1.45521 - 0.210395i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-297. + 416. i)T \)
good3 \( 1 + 2.59e4T + 3.87e8T^{2} \)
5 \( 1 - 1.90e6iT - 3.81e12T^{2} \)
7 \( 1 - 9.33e6iT - 1.62e15T^{2} \)
11 \( 1 - 2.94e9T + 5.55e18T^{2} \)
13 \( 1 + 7.67e9iT - 1.12e20T^{2} \)
17 \( 1 + 1.40e10T + 1.40e22T^{2} \)
19 \( 1 - 5.59e11T + 1.04e23T^{2} \)
23 \( 1 - 2.61e12iT - 3.24e24T^{2} \)
29 \( 1 + 1.91e13iT - 2.10e26T^{2} \)
31 \( 1 - 3.57e13iT - 6.99e26T^{2} \)
37 \( 1 - 2.51e14iT - 1.68e28T^{2} \)
41 \( 1 - 7.94e13T + 1.07e29T^{2} \)
43 \( 1 + 3.62e14T + 2.52e29T^{2} \)
47 \( 1 + 2.20e14iT - 1.25e30T^{2} \)
53 \( 1 - 1.22e15iT - 1.08e31T^{2} \)
59 \( 1 - 2.66e15T + 7.50e31T^{2} \)
61 \( 1 + 2.02e16iT - 1.36e32T^{2} \)
67 \( 1 - 3.97e16T + 7.40e32T^{2} \)
71 \( 1 + 1.25e16iT - 2.10e33T^{2} \)
73 \( 1 - 6.29e15T + 3.46e33T^{2} \)
79 \( 1 - 6.53e16iT - 1.43e34T^{2} \)
83 \( 1 - 1.09e17T + 3.49e34T^{2} \)
89 \( 1 + 3.14e17T + 1.22e35T^{2} \)
97 \( 1 + 5.54e17T + 5.77e35T^{2} \)
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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

−17.44298608354067781236448293447, −15.43748065701116826457016989369, −13.91165048342826418290050171449, −11.99608350902543324609138878767, −11.26796881720380996906952701614, −9.870160987630335420270923633952, −6.57595996302880613573781802077, −5.29345763252019602704798081606, −3.28423586349785818691952250609, −1.08211383900967262149149125038, 0.74656490240077203209065515775, 4.25968940074835591751080093010, 5.50382863404481145020222381099, 6.89269041380690275825763022934, 9.028670039137924168588751575557, 11.57187972950771222353969881416, 12.55794038138931590542901979668, 14.25417141907341205119005754918, 16.35806505623895132148519048638, 16.72091730765816021787088908069

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