Properties

Degree 2
Conductor $ 2^{7} $
Sign $i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.79·3-s − 8i·5-s − 78.3i·7-s + 14.9·9-s − 107.·11-s − 216i·13-s − 78.3i·15-s − 162·17-s + 440.·19-s − 767. i·21-s + 705. i·23-s + 561·25-s − 646.·27-s − 1.30e3i·29-s + 627. i·31-s + ⋯
L(s)  = 1  + 1.08·3-s − 0.320i·5-s − 1.59i·7-s + 0.185·9-s − 0.890·11-s − 1.27i·13-s − 0.348i·15-s − 0.560·17-s + 1.22·19-s − 1.74i·21-s + 1.33i·23-s + 0.897·25-s − 0.887·27-s − 1.55i·29-s + 0.652i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(128\)    =    \(2^{7}\)
\( \varepsilon \)  =  $i$
motivic weight  =  \(4\)
character  :  $\chi_{128} (63, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 128,\ (\ :2),\ i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.51483 - 1.51483i\)
\(L(\frac12)\)  \(\approx\)  \(1.51483 - 1.51483i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 9.79T + 81T^{2} \)
5 \( 1 + 8iT - 625T^{2} \)
7 \( 1 + 78.3iT - 2.40e3T^{2} \)
11 \( 1 + 107.T + 1.46e4T^{2} \)
13 \( 1 + 216iT - 2.85e4T^{2} \)
17 \( 1 + 162T + 8.35e4T^{2} \)
19 \( 1 - 440.T + 1.30e5T^{2} \)
23 \( 1 - 705. iT - 2.79e5T^{2} \)
29 \( 1 + 1.30e3iT - 7.07e5T^{2} \)
31 \( 1 - 627. iT - 9.23e5T^{2} \)
37 \( 1 + 1.51e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.89e3T + 2.82e6T^{2} \)
43 \( 1 - 2.90e3T + 3.41e6T^{2} \)
47 \( 1 + 1.41e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.97e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.26e3T + 1.21e7T^{2} \)
61 \( 1 - 2.37e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.67e3T + 2.01e7T^{2} \)
71 \( 1 - 7.75e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.75e3T + 2.83e7T^{2} \)
79 \( 1 - 7.99e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.33e3T + 4.74e7T^{2} \)
89 \( 1 - 2.43e3T + 6.27e7T^{2} \)
97 \( 1 - 7.45e3T + 8.85e7T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.78649603616199776297497775814, −11.13595386173082473098781621634, −10.22086837550269139758518234111, −9.191713643261519523946890438508, −7.84752100203802166133265794214, −7.46300456700746329920712342331, −5.48470306683953551851014093124, −3.96731406971884935726124658095, −2.79943439299283663261414756601, −0.78513212578891077436934625778, 2.23060336465473971407639389427, 3.02782882218380617815054436910, 4.90759398588364198377024704612, 6.31301053656432889173589406297, 7.72900218503414445701084150518, 8.868735491195685769555493975029, 9.286203370614414740929098528466, 10.86043022133786562387814191846, 11.96150707873576711649320554230, 12.94467264018292801163163508469

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