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Properties

Label 2-1512-8.5-c1-0-46
Degree $2$
Conductor $1512$
Sign $0.472 - 0.881i$
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  + (0.497 + 1.32i)2-s + (−1.50 + 1.31i)4-s − 1.25i·5-s + 7-s + (−2.49 − 1.33i)8-s + (1.65 − 0.624i)10-s − 1.55i·11-s + 1.07i·13-s + (0.497 + 1.32i)14-s + (0.528 − 3.96i)16-s + 0.0158·17-s + 2.35i·19-s + (1.65 + 1.88i)20-s + (2.06 − 0.775i)22-s + 5.95·23-s + ⋯
L(s)  = 1  + (0.351 + 0.936i)2-s + (−0.752 + 0.658i)4-s − 0.560i·5-s + 0.377·7-s + (−0.881 − 0.472i)8-s + (0.524 − 0.197i)10-s − 0.469i·11-s + 0.297i·13-s + (0.133 + 0.353i)14-s + (0.132 − 0.991i)16-s + 0.00383·17-s + 0.539i·19-s + (0.369 + 0.421i)20-s + (0.439 − 0.165i)22-s + 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.937169301\)
\(L(\frac12)\) \(\approx\) \(1.937169301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.497 - 1.32i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.25iT - 5T^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
13 \( 1 - 1.07iT - 13T^{2} \)
17 \( 1 - 0.0158T + 17T^{2} \)
19 \( 1 - 2.35iT - 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 + 0.469iT - 29T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 - 4.59iT - 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + 1.97iT - 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 + 1.86iT - 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 - 3.92iT - 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 4.22T + 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 + 8.88iT - 83T^{2} \)
89 \( 1 + 0.240T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−9.194068424852696993441778434516, −8.757278751518251929806823831233, −7.965181475313513311127001298871, −7.19835990629191644998485256200, −6.31429198564815188617484064941, −5.47315419668786012206945415593, −4.75160917051390578272831458805, −3.92584320305692319049559943886, −2.77651579076203750054839567300, −0.987187092972617374714131237451, 0.972475048940122119393343199632, 2.34931589217680610556810972879, 3.07740763956705648319826823714, 4.21478968576199596376086707794, 4.96995429022219438818236732770, 5.86900729770929159761925014397, 6.88918294538902403643225114284, 7.73466816838107609678760853923, 8.866125473792213025087459690925, 9.380270868469511840023450424102

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