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Properties

Label 2-3072-96.77-c0-0-3
Degree $2$
Conductor $3072$
Sign $-0.195 + 0.980i$
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

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-0.195 + 0.980i$
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: \((2,\ 3072,\ (\ :0),\ -0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07892344976\)
\(L(\frac12)\) \(\approx\) \(0.07892344976\)
\(L(1)\) 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

−8.800199390979243946567472364596, −8.297480074110232768846406169110, −6.69483648461037108950011381480, −6.53338337997101230124907939853, −5.48133845749892231996230538732, −4.98365466667422864926448008488, −3.93749432041398216757598256349, −3.01485152115033405575705328468, −2.37346222347245491448588230473, −0.05041344215138490145195803139, 1.32973211895075168880117986978, 2.55093837591814356978759055346, 3.51262110243495553771533600252, 4.41874878549853194769204215764, 5.39897859217310411951579101122, 6.47100643543446462662721670899, 6.71947711446569707399101307761, 7.41586875574753035675480203817, 8.171619339778313770048525998767, 9.086211902948723513884784772043

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