Properties

Label 2-3072-48.29-c0-0-7
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  i·3-s − 9-s + (1 + i)11-s − 2i·17-s + (1 + i)19-s i·25-s + i·27-s + (1 − i)33-s + (1 − i)43-s + 49-s − 2·51-s + (1 − i)57-s + (−1 − i)59-s + (1 + i)67-s − 75-s + ⋯
L(s)  = 1  i·3-s − 9-s + (1 + i)11-s − 2i·17-s + (1 + i)19-s i·25-s + i·27-s + (1 − i)33-s + (1 − i)43-s + 49-s − 2·51-s + (1 − i)57-s + (−1 − i)59-s + (1 + i)67-s − 75-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

\(L(\frac{1}{2})\) \(\approx\) \(1.287371027\)
\(L(\frac12)\) \(\approx\) \(1.287371027\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (1 + i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.742067011689231100751771517544, −7.77401081529444014647453021829, −7.23648074327990459567691904638, −6.69901979833410237389365337597, −5.78976522458216297396938701614, −5.01203155720588031419351183405, −4.00970815369222455794927578997, −2.92104513493754761761269315478, −2.04257829060276103367834255572, −0.943485650547084277800167327003, 1.27070070165562505454868999107, 2.78031566266009699371606239922, 3.64573613087210841047816004284, 4.16425258856899897415220870293, 5.23607029821142325538584616136, 5.91338130275833326755794402507, 6.56792198876660692623188299498, 7.69609808001593294661404565500, 8.480720198105445650540411152298, 9.120861765805167954778907475914

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