Properties

Degree $2$
Conductor $128$
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

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-i$
Motivic weight: \(4\)
Character: $\chi_{128} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.51483 + 1.51483i\)
\(L(\frac12)\) \(\approx\) \(1.51483 + 1.51483i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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