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Properties

Label 2-3072-96.53-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  + (0.923 − 0.382i)3-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)9-s + (0.292 + 0.707i)13-s + (−0.541 − 1.30i)19-s + (−1.70 − 0.707i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s − 0.765·31-s + (0.707 − 1.70i)37-s + (0.541 + 0.541i)39-s + (1.30 + 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (0.707 − 0.292i)61-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)9-s + (0.292 + 0.707i)13-s + (−0.541 − 1.30i)19-s + (−1.70 − 0.707i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s − 0.765·31-s + (0.707 − 1.70i)37-s + (0.541 + 0.541i)39-s + (1.30 + 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (0.707 − 0.292i)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: $\chi_{3072} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ -0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345114875\)
\(L(\frac12)\) \(\approx\) \(1.345114875\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
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 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + 0.765T + T^{2} \)
37 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
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

−8.833103453447186263727447332752, −7.77597920838436434580234729093, −7.17378365087567381288122211045, −6.66202056633906106514807343567, −5.92618861971139687118850252548, −4.31076170645357010528858101792, −3.99849241713652084948564230297, −3.03634823113941890618423380034, −2.14296035011328333502575124918, −0.71570560363981475023299691122, 1.81251205300347804146669034936, 2.78796803851887883902450408014, 3.38655850740737544284869700820, 4.18886578508731061200869134158, 5.47817065783912085106854692279, 5.92076279634738472581804482089, 6.86091299988855281049554553750, 7.84141960585284035120285430220, 8.444660615322976487634228736965, 9.104850391077995374674973817896

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