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Properties

Label 2-3072-48.29-c0-0-2
Degree $2$
Conductor $3072$
Sign $0.382 - 0.923i$
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  + (−0.707 + 0.707i)3-s − 1.00i·9-s + (1.41 + 1.41i)11-s − i·25-s + (0.707 + 0.707i)27-s − 2.00·33-s + 49-s + (1.41 + 1.41i)59-s + 2i·73-s + (0.707 + 0.707i)75-s − 1.00·81-s + (−1.41 + 1.41i)83-s + 2·97-s + (1.41 − 1.41i)99-s + (−1.41 − 1.41i)107-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 1.00i·9-s + (1.41 + 1.41i)11-s − i·25-s + (0.707 + 0.707i)27-s − 2.00·33-s + 49-s + (1.41 + 1.41i)59-s + 2i·73-s + (0.707 + 0.707i)75-s − 1.00·81-s + (−1.41 + 1.41i)83-s + 2·97-s + (1.41 − 1.41i)99-s + (−1.41 − 1.41i)107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038388125\)
\(L(\frac12)\) \(\approx\) \(1.038388125\)
\(L(1)\) 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 \( 1 + iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−9.143587767892117057467546409334, −8.501876797379283641808797221342, −7.26193297399623830351969051673, −6.75695514858463243316279194096, −5.99574931940502672455653576533, −5.12663337476553957180311472218, −4.24438376670352310000488130924, −3.90294273121176848437951031786, −2.50229553473516403073351853980, −1.21653681652877569593334382239, 0.848209316173399195777955746979, 1.83951326644066734521503688133, 3.15747861379236883038058783961, 4.00689566264957394810232429056, 5.10259930224557558556830228620, 5.86135806230783521844654220983, 6.43158330850156362504776652469, 7.11794341635796026076304256279, 7.943758493597920639692317887076, 8.732914256296874498474520156530

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