Properties

Label 2-3072-8.5-c1-0-8
Degree $2$
Conductor $3072$
Sign $-1$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
Motivic weight $1$
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 + 1.41i·5-s − 9-s + 4i·11-s + 1.41i·13-s − 1.41·15-s − 4i·19-s − 5.65·23-s + 2.99·25-s i·27-s + 7.07i·29-s + 5.65·31-s − 4·33-s + 4.24i·37-s − 1.41·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.632i·5-s − 0.333·9-s + 1.20i·11-s + 0.392i·13-s − 0.365·15-s − 0.917i·19-s − 1.17·23-s + 0.599·25-s − 0.192i·27-s + 1.31i·29-s + 1.01·31-s − 0.696·33-s + 0.697i·37-s − 0.226·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-1$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026882244\)
\(L(\frac12)\) \(\approx\) \(1.026882244\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.232126814578320429813384818197, −8.339275940166700627558172399414, −7.58290350370398758733335000716, −6.67904545383706268775987681047, −6.30156111303892850483870953887, −4.81994935622962626634728503357, −4.72211497213393291118541848319, −3.47501187046849021814280238909, −2.71911860095142834955127902418, −1.62755909088188818399364015141, 0.31952836919656524250667600605, 1.37516253127182251074843668782, 2.51619090944501695618862619177, 3.52494178942616757994885140449, 4.38952947145505462266646511097, 5.53095463387614374824168242353, 5.92910193474162116281494105477, 6.77774119109591027452095489305, 7.80744284128429332916174100518, 8.308358390561648159039915870940

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