Properties

Label 2-3072-8.5-c1-0-60
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 − 3.16i·5-s − 1.74·7-s − 9-s − 6.47i·11-s − 1.41i·13-s + 3.16·15-s + 2.47·17-s − 6.47i·19-s − 1.74i·21-s − 5.65·23-s − 5.00·25-s i·27-s + 5.99i·29-s − 3.90·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.41i·5-s − 0.660·7-s − 0.333·9-s − 1.95i·11-s − 0.392i·13-s + 0.816·15-s + 0.599·17-s − 1.48i·19-s − 0.381i·21-s − 1.17·23-s − 1.00·25-s − 0.192i·27-s + 1.11i·29-s − 0.702·31-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

\(L(1)\) \(\approx\) \(0.6976957202\)
\(L(\frac12)\) \(\approx\) \(0.6976957202\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.16iT - 5T^{2} \)
7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 + 6.47iT - 11T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 5.99iT - 29T^{2} \)
31 \( 1 + 3.90T + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + 1.52iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + 8.94iT - 59T^{2} \)
61 \( 1 + 2.08iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 9.15T + 71T^{2} \)
73 \( 1 + 2.94T + 73T^{2} \)
79 \( 1 + 7.40T + 79T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.509531716477298687614340140746, −7.905832768982935243087815081357, −6.66771606306216593776238511253, −5.80589077682777862044325693898, −5.30936909889714202292474239669, −4.48977032521143098021116983329, −3.51068901669512462757998496729, −2.88513157761559430370994188929, −1.18262454012796432259325275961, −0.22602559747899664078648198914, 1.84925629362466139562175140718, 2.37151796736590827690585381302, 3.54260182105689786557728577656, 4.15782555018239913203636993585, 5.52169981088562581786623770677, 6.28362518875561054514441719608, 6.82784229746110291889110560219, 7.57690987946124431182936326606, 7.898302499050237591611998985814, 9.302942851871403501991839118496

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