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Properties

Label 2-1512-9.7-c1-0-6
Degree $2$
Conductor $1512$
Sign $0.415 - 0.909i$
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  + (1.21 + 2.10i)5-s + (−0.5 + 0.866i)7-s + (0.379 − 0.657i)11-s + (−1.11 − 1.92i)13-s + 7.04·17-s − 1.37·19-s + (3.51 + 6.09i)23-s + (−0.467 + 0.810i)25-s + (0.418 − 0.724i)29-s + (0.265 + 0.459i)31-s − 2.43·35-s + 4.53·37-s + (4.42 + 7.67i)41-s + (−3.70 + 6.41i)43-s + (−3.39 + 5.88i)47-s + ⋯
L(s)  = 1  + (0.544 + 0.943i)5-s + (−0.188 + 0.327i)7-s + (0.114 − 0.198i)11-s + (−0.309 − 0.535i)13-s + 1.70·17-s − 0.314·19-s + (0.733 + 1.27i)23-s + (−0.0935 + 0.162i)25-s + (0.0776 − 0.134i)29-s + (0.0476 + 0.0825i)31-s − 0.411·35-s + 0.744·37-s + (0.691 + 1.19i)41-s + (−0.564 + 0.977i)43-s + (−0.495 + 0.858i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836707945\)
\(L(\frac12)\) \(\approx\) \(1.836707945\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.21 - 2.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.379 + 0.657i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.11 + 1.92i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.04T + 17T^{2} \)
19 \( 1 + 1.37T + 19T^{2} \)
23 \( 1 + (-3.51 - 6.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.418 + 0.724i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.265 - 0.459i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + (-4.42 - 7.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.70 - 6.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.39 - 5.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.607T + 53T^{2} \)
59 \( 1 + (0.581 + 1.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.85 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.152 - 0.264i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (7.62 - 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.18 + 14.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + (-5.46 + 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.866326387838091644960647159570, −8.924973059589130722350152305872, −7.83923823201032532122590416221, −7.30334642479453809158678857608, −6.16039482074604001627570284105, −5.78108788365695298264979782865, −4.67326844307904687867047276785, −3.24137943801185898614230691495, −2.82587186032380850409672705213, −1.34273902263531724511882366959, 0.809447279753490207202708999874, 1.97148874326865594975696996189, 3.27752032399634203550236798517, 4.40194327389938239158674771850, 5.13790567350834360390102811145, 5.97354437123287126599092292662, 6.92306393404218699582198153979, 7.73733728649963558779017438379, 8.704959712117486334463094027071, 9.265170193324531439147249653451

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