Properties

Label 2-3072-48.5-c0-0-5
Degree $2$
Conductor $3072$
Sign $0.923 + 0.382i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
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  − 3-s + 9-s + (−1 + i)11-s − 2i·17-s + (1 − i)19-s + i·25-s − 27-s + (1 − i)33-s + (1 + i)43-s + 49-s + 2i·51-s + (−1 + i)57-s + (1 − i)59-s + (1 − i)67-s i·75-s + ⋯
L(s)  = 1  − 3-s + 9-s + (−1 + i)11-s − 2i·17-s + (1 − i)19-s + i·25-s − 27-s + (1 − i)33-s + (1 + i)43-s + 49-s + 2i·51-s + (−1 + i)57-s + (1 − i)59-s + (1 − i)67-s i·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8255054004\)
\(L(\frac12)\) \(\approx\) \(0.8255054004\)
\(L(1)\) 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 \( 1 + T \)
good5 \( 1 - iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−9.134789036095722181706723046940, −7.69989132846954318099785318847, −7.32920848998475293440584309054, −6.72810762523987802388296506861, −5.56503528388875144518949146147, −5.05058586032667330641143338617, −4.52430033146478063012974348020, −3.17825728774416267984895776629, −2.21222672955083138994476016064, −0.75825816710647015445463612734, 0.988624387561853365470562512342, 2.25057310954275461927680389280, 3.57151940899356055238328739902, 4.22964232404430460503467035321, 5.45892221723750339750860124333, 5.73292479242823631303042864156, 6.47537485738457200164339407308, 7.47583931653308452568269025400, 8.126544118231910473478541616356, 8.803168269050532159351772513753

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