Properties

Label 2-3072-48.5-c0-0-9
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

\(L(\frac{1}{2})\) \(\approx\) \(1.883387391\)
\(L(\frac12)\) \(\approx\) \(1.883387391\)
\(L(1)\) 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

−8.991867241591200503274452823422, −8.187908413811028477961657965379, −7.38439463536526416493683727925, −6.76716017620525827439057530610, −5.85858686958741106402975018290, −4.87392893941052359135438913999, −3.90887775968505128405381294931, −3.29686819845482800439453457706, −2.34616180650914776357796903068, −1.19374793950770575579742669565, 1.56971698592961988803804701966, 2.25214638278143473372484537808, 3.42225074320931237966468349963, 4.22967984828198272831790078940, 4.71392230783152933257892793228, 6.27379427249793469489236316305, 6.59974280053884547985669621562, 7.57835301984091526885560966224, 8.278518913699956095155232089547, 8.889169388916295252295546034950

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