Properties

Label 2-2e7-8.5-c1-0-1
Degree $2$
Conductor $128$
Sign $0.707 - 0.707i$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·5-s + 3·9-s − 4i·13-s − 2·17-s − 11·25-s − 4i·29-s − 12i·37-s + 10·41-s + 12i·45-s − 7·49-s + 4i·53-s + 12i·61-s + 16·65-s + 6·73-s + 9·81-s + ⋯
L(s)  = 1  + 1.78i·5-s + 9-s − 1.10i·13-s − 0.485·17-s − 2.20·25-s − 0.742i·29-s − 1.97i·37-s + 1.56·41-s + 1.78i·45-s − 49-s + 0.549i·53-s + 1.53i·61-s + 1.98·65-s + 0.702·73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01852 + 0.421885i\)
\(L(\frac12)\) \(\approx\) \(1.01852 + 0.421885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 3T^{2} \)
5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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

−13.53783374545753250377939509110, −12.51293411533616317032473602234, −11.11911838669299626302245413709, −10.51740556902965316698962093181, −9.576316991150715064482865075460, −7.79814902851498306451719208502, −7.01704861305086744721648990389, −5.89147895045870161977741553862, −3.97199102139077686237606533667, −2.57482920414217144195647823211, 1.52019857626457482641329780176, 4.20531702035781344901108809042, 5.01237511108207605464128768470, 6.62189904477336838247127256571, 8.036557796973512610753317860725, 9.061941896594789261995051039125, 9.803680106827546855846298258804, 11.36090455256956921638940592955, 12.41930941597087059874329181038, 13.02824755955478473580174590461

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