Properties

Label 2-1536-3.2-c0-0-1
Degree $2$
Conductor $1536$
Sign $0.707 - 0.707i$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + 1.00i·9-s + 1.41i·11-s + 2i·17-s − 1.41·19-s + 25-s + (0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41·43-s − 49-s + (1.41 − 1.41i)51-s + (1.00 + 1.00i)57-s + 1.41i·59-s + 1.41·67-s + (−0.707 − 0.707i)75-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + 1.00i·9-s + 1.41i·11-s + 2i·17-s − 1.41·19-s + 25-s + (0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41·43-s − 49-s + (1.41 − 1.41i)51-s + (1.00 + 1.00i)57-s + 1.41i·59-s + 1.41·67-s + (−0.707 − 0.707i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7510711367\)
\(L(\frac12)\) \(\approx\) \(0.7510711367\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.971116130565724134157014153831, −8.777771330674611926463490035241, −8.088559094822749314830387147590, −7.21641884535621143701813114205, −6.51550777352264895345966044221, −5.84504045857925779397450540147, −4.75180123181174105061516764535, −4.04833588597752672251228191017, −2.38832022748634222207639775930, −1.53603653617730186155043298715, 0.68119417147713322337564506440, 2.68248220446785032416315016012, 3.61448118192798378442395823001, 4.66686379699381178383873813382, 5.34020912141235155753531105717, 6.26068990860901136944665667563, 6.89078908662822911633680426017, 8.083274555376128056894866359870, 8.959691317762781180466982587569, 9.475815298205732827349826490842

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