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Properties

Label 2-3072-96.77-c0-0-1
Degree $2$
Conductor $3072$
Sign $0.980 + 0.195i$
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  + (0.382 − 0.923i)3-s + (−1.30 + 1.30i)7-s + (−0.707 − 0.707i)9-s + (0.707 + 0.292i)13-s + (1.30 + 0.541i)19-s + (0.707 + 1.70i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 0.765·31-s + (1.70 − 0.707i)37-s + (0.541 − 0.541i)39-s + (0.541 + 1.30i)43-s − 2.41i·49-s + (1 − 0.999i)57-s + (−0.292 + 0.707i)61-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−1.30 + 1.30i)7-s + (−0.707 − 0.707i)9-s + (0.707 + 0.292i)13-s + (1.30 + 0.541i)19-s + (0.707 + 1.70i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 0.765·31-s + (1.70 − 0.707i)37-s + (0.541 − 0.541i)39-s + (0.541 + 1.30i)43-s − 2.41i·49-s + (1 − 0.999i)57-s + (−0.292 + 0.707i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262191598\)
\(L(\frac12)\) \(\approx\) \(1.262191598\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - 0.765T + T^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + 0.765iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−8.881537155682345162585936377141, −8.148270098366385646942605730118, −7.39720450061855789651107389653, −6.31859542662500620192176360532, −6.21728984205062153649989510985, −5.30327738349600097731930540059, −3.91305067472015017011031558809, −2.95839065156488726153162545013, −2.49832768082898656200248760424, −1.12187665081431901841130386061, 0.931215253164968913758537572090, 2.79884772833440522923022029207, 3.37583151697257729870363791436, 4.05944972212668805903216496299, 4.91661853847750806886652219655, 5.85882162429149941180211201327, 6.69842300400085470818056689438, 7.44552261218421021875764502144, 8.186945964829954417709176992773, 9.186379910839255918609585796266

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