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Properties

Label 2-3072-48.29-c0-0-10
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

\(L(\frac{1}{2})\) \(\approx\) \(1.284489029\)
\(L(\frac12)\) \(\approx\) \(1.284489029\)
\(L(1)\) 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

−8.486155238957670504644861026526, −7.992583294387787936435018286802, −7.35396265831435034975382312487, −6.35987649485421005494787342589, −5.79916754763116127285146233624, −4.82184414023189294510672141893, −3.63967306527139709749155052014, −2.91612516622658256786460100480, −2.14011945446793410127392067925, −0.68821966455674584709200266864, 1.86676712152264104142063758650, 2.65376650948268390370063819365, 3.54815237099680412111304605985, 4.57924482306293120338808032144, 5.02716164288572057464889281164, 5.93490095436040219595301790049, 7.30127145106638879911113868013, 7.53679430880026317119010240868, 8.409048792057003931180090058650, 9.221599836981437949538620642462

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