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Properties

Label 2-3072-96.5-c0-0-7
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 + (−0.541 − 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (1.30 − 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s − 1.84·31-s + (−0.707 − 0.292i)37-s + (−1.30 − 1.30i)39-s + (0.541 − 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)61-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (−0.541 − 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (1.30 − 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s − 1.84·31-s + (−0.707 − 0.292i)37-s + (−1.30 − 1.30i)39-s + (0.541 − 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)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: $\chi_{3072} (2945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025832359\)
\(L(\frac12)\) \(\approx\) \(1.025832359\)
\(L(1)\) 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 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + 1.84T + T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + 1.84iT - T^{2} \)
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

−8.712706514979178064166189153602, −7.68254739435718214780128024104, −7.22198492336074479907750216967, −6.48245369677368790700035975949, −5.68282358977591862882424819219, −5.12777019075454864932852729702, −3.66951290686587161244252711823, −3.16539024848146181297138486773, −1.75808958739044999023835972064, −0.74707294803367714676339572193, 1.36232259967031122454416315607, 2.91434562437788430552093691004, 3.60372134981667828154407606084, 4.36081702301021521311821114582, 5.39734321440826482785734564183, 5.96920594883192015153989678921, 6.58623628192296696979889093152, 7.63436285226744109892824272142, 8.755645887055584456359582255819, 9.012308978222314166221762016416

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