↑

Properties

Label 2-2e7-16.13-c1-0-1
Degree $2$
Conductor $128$
Sign $0.382 + 0.923i$
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  + (−1 − i)3-s + (1 − i)5-s − 2i·7-s − i·9-s + (1 − i)11-s + (1 + i)13-s − 2·15-s − 2·17-s + (3 + 3i)19-s + (−2 + 2i)21-s + 6i·23-s + 3i·25-s + (−4 + 4i)27-s + (−3 − 3i)29-s + 8·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.447 − 0.447i)5-s − 0.755i·7-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (0.277 + 0.277i)13-s − 0.516·15-s − 0.485·17-s + (0.688 + 0.688i)19-s + (−0.436 + 0.436i)21-s + 1.25i·23-s + 0.600i·25-s + (−0.769 + 0.769i)27-s + (−0.557 − 0.557i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799819 - 0.534422i\)
\(L(\frac12)\) \(\approx\) \(0.799819 - 0.534422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 3iT^{2} \)
5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (3 - 3i)T - 59iT^{2} \)
61 \( 1 + (-9 - 9i)T + 61iT^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−13.30409326191450323749036487387, −12.04703488585734106578020538026, −11.34988060586784876578537077178, −10.01906219176728977678061627474, −9.013444924239594305077968375781, −7.57636266886867046204462624117, −6.49880138617070707658762768363, −5.44908684486335744558365099104, −3.78289150363022985981756278865, −1.29307975640473051161799074505, 2.56170305732102056983156404427, 4.53151267740980108681903213984, 5.66975054063335894590245814235, 6.75957304751256734756669518283, 8.352243291068938215378300579181, 9.544068541528414162031497236083, 10.50284001106027731166125190930, 11.33777772939555990236655141902, 12.39725935628457991775816094547, 13.57763943423367984700554367560

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