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Properties

Label 2-1512-24.11-c1-0-3
Degree $2$
Conductor $1512$
Sign $-0.505 + 0.863i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (0.481 + 1.32i)2-s + (−1.53 + 1.27i)4-s + 0.436·5-s + i·7-s + (−2.44 − 1.42i)8-s + (0.209 + 0.580i)10-s − 2.73i·11-s − 1.64i·13-s + (−1.32 + 0.481i)14-s + (0.725 − 3.93i)16-s + 5.22i·17-s − 5.99·19-s + (−0.670 + 0.558i)20-s + (3.64 − 1.31i)22-s − 7.68·23-s + ⋯
L(s)  = 1  + (0.340 + 0.940i)2-s + (−0.768 + 0.639i)4-s + 0.195·5-s + 0.377i·7-s + (−0.863 − 0.505i)8-s + (0.0663 + 0.183i)10-s − 0.825i·11-s − 0.456i·13-s + (−0.355 + 0.128i)14-s + (0.181 − 0.983i)16-s + 1.26i·17-s − 1.37·19-s + (−0.149 + 0.124i)20-s + (0.776 − 0.280i)22-s − 1.60·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.505 + 0.863i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.505 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3003628521\)
\(L(\frac12)\) \(\approx\) \(0.3003628521\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.481 - 1.32i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.436T + 5T^{2} \)
11 \( 1 + 2.73iT - 11T^{2} \)
13 \( 1 + 1.64iT - 13T^{2} \)
17 \( 1 - 5.22iT - 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 - 9.77iT - 31T^{2} \)
37 \( 1 - 3.81iT - 37T^{2} \)
41 \( 1 + 9.74iT - 41T^{2} \)
43 \( 1 + 8.84T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 - 6.26iT - 61T^{2} \)
67 \( 1 - 9.42T + 67T^{2} \)
71 \( 1 + 9.76T + 71T^{2} \)
73 \( 1 + 9.14T + 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 - 0.227iT - 83T^{2} \)
89 \( 1 + 2.46iT - 89T^{2} \)
97 \( 1 + 3.37T + 97T^{2} \)
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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

−9.986157555169590122224363015396, −8.646466836506665972102670517560, −8.570199273611412442270558334095, −7.63652305002718806701040936651, −6.55099651670619747657209417191, −5.94667820282022741110009868203, −5.34571742824217061298327461774, −4.11061829265513914424719265378, −3.45101933007168216334572111167, −2.00818328977544162221389016166, 0.097796670479734469423254465185, 1.81210748222527380496260286254, 2.50916996716214741503216244896, 4.00566575050182932687180575945, 4.33829337532150495117365508843, 5.50745224991745020361562211423, 6.30630993891748154173072580533, 7.36216635525423698638106243449, 8.261760490011444089103328378795, 9.324132965068059809116370103405

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