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Properties

Label 2-3072-48.29-c0-0-11
Degree $2$
Conductor $3072$
Sign $-0.923 + 0.382i$
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  + (0.707 − 0.707i)3-s + (−1 − i)5-s − 1.41i·7-s − 1.00i·9-s − 1.41·15-s + (−1.00 − 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (−1 + i)29-s + 1.41·31-s + (−1.41 + 1.41i)35-s + (−1.00 + 1.00i)45-s − 1.00·49-s + (−1 − i)53-s + (1.41 + 1.41i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−1 − i)5-s − 1.41i·7-s − 1.00i·9-s − 1.41·15-s + (−1.00 − 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (−1 + i)29-s + 1.41·31-s + (−1.41 + 1.41i)35-s + (−1.00 + 1.00i)45-s − 1.00·49-s + (−1 − i)53-s + (1.41 + 1.41i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-0.923 + 0.382i$
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: \((2,\ 3072,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150012530\)
\(L(\frac12)\) \(\approx\) \(1.150012530\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (1 + i)T + iT^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.266606216722696494091143372944, −7.983199377571037643222278914115, −7.15318769952046467360592223629, −6.69021633211812480094545910675, −5.41189634486789386131586654483, −4.36507057913104450224794276878, −3.91115950058971451499093353186, −3.01767214057854747039064491074, −1.56212013498722695407897813421, −0.66157097547778919428194430940, 2.17195891256597214698464822127, 2.83999079541757872408194673121, 3.58028062500165148710282559752, 4.39712775542095332692234536015, 5.32714922457050951722572177407, 6.19135945836278037402085234544, 7.09118609714977698830785941425, 8.034731651387057405663089212255, 8.273109861747792222942724046039, 9.280963923865094030358655390498

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