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Properties

Label 2-147-7.5-c6-0-7
Degree $2$
Conductor $147$
Sign $-0.553 - 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.80 − 3.12i)2-s + (−13.5 + 7.79i)3-s + (25.4 + 44.1i)4-s + (−71.9 − 41.5i)5-s + 56.2i·6-s + 414.·8-s + (121.5 − 210. i)9-s + (−259. + 149. i)10-s + (221. + 383. i)11-s + (−688. − 397. i)12-s + 696. i·13-s + 1.29e3·15-s + (−883. + 1.53e3i)16-s + (5.44e3 − 3.14e3i)17-s + (−438. − 758. i)18-s + (2.32e3 + 1.34e3i)19-s + ⋯
L(s)  = 1  + (0.225 − 0.390i)2-s + (−0.5 + 0.288i)3-s + (0.398 + 0.690i)4-s + (−0.575 − 0.332i)5-s + 0.260i·6-s + 0.810·8-s + (0.166 − 0.288i)9-s + (−0.259 + 0.149i)10-s + (0.166 + 0.287i)11-s + (−0.398 − 0.230i)12-s + 0.317i·13-s + 0.383·15-s + (−0.215 + 0.373i)16-s + (1.10 − 0.640i)17-s + (−0.0751 − 0.130i)18-s + (0.339 + 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.553 - 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.094366407\)
\(L(\frac12)\) \(\approx\) \(1.094366407\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-1.80 + 3.12i)T + (-32 - 55.4i)T^{2} \)
5 \( 1 + (71.9 + 41.5i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-221. - 383. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 696. iT - 4.82e6T^{2} \)
17 \( 1 + (-5.44e3 + 3.14e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-2.32e3 - 1.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (7.88e3 - 1.36e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 2.32e4T + 5.94e8T^{2} \)
31 \( 1 + (4.11e4 - 2.37e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-5.07e3 + 8.79e3i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 3.81e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.51e5T + 6.32e9T^{2} \)
47 \( 1 + (-4.35e4 - 2.51e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-9.97e4 - 1.72e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (3.35e5 - 1.93e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.02e4 - 5.89e3i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.92e5 - 3.32e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 1.56e5T + 1.28e11T^{2} \)
73 \( 1 + (3.25e5 - 1.87e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.65e4 - 2.87e4i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 9.84e5iT - 3.26e11T^{2} \)
89 \( 1 + (-2.27e5 - 1.31e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 5.75e5iT - 8.32e11T^{2} \)
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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

−11.92875869437796186579123513207, −11.68713455458063482272029577979, −10.45728234970314610104405917370, −9.317299684082695325752434825868, −7.908158362867372607604948065735, −7.12605539056569777535242568275, −5.54310732764139300431432472568, −4.24782319660920900322819209227, −3.30113759255934728066338987884, −1.53305366681238357444478499131, 0.32130604628070188230093502892, 1.79186203132370454970928035520, 3.65810348472996861308097978455, 5.22584361300611213330014141652, 6.10229264834451303234278000692, 7.16785937836099109414134502031, 8.052055787671752369738468499139, 9.752601436710210519289425605748, 10.75348340401240717096376677034, 11.47580987466115351475943387221

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