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Properties

Label 2-147-21.20-c3-0-28
Degree $2$
Conductor $147$
Sign $-0.943 + 0.330i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.20i·2-s + (0.930 + 5.11i)3-s − 2.24·4-s − 11.9·5-s + (16.3 − 2.97i)6-s − 18.4i·8-s + (−25.2 + 9.50i)9-s + 38.3i·10-s − 60.3i·11-s + (−2.08 − 11.4i)12-s − 3.46i·13-s + (−11.1 − 61.2i)15-s − 76.9·16-s − 32.4·17-s + (30.4 + 80.8i)18-s − 142. i·19-s + ⋯
L(s)  = 1  − 1.13i·2-s + (0.178 + 0.983i)3-s − 0.280·4-s − 1.07·5-s + (1.11 − 0.202i)6-s − 0.814i·8-s + (−0.935 + 0.352i)9-s + 1.21i·10-s − 1.65i·11-s + (−0.0501 − 0.275i)12-s − 0.0738i·13-s + (−0.191 − 1.05i)15-s − 1.20·16-s − 0.463·17-s + (0.398 + 1.05i)18-s − 1.72i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ -0.943 + 0.330i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.147976 - 0.869434i\)
\(L(\frac12)\) \(\approx\) \(0.147976 - 0.869434i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.930 - 5.11i)T \)
7 \( 1 \)
good2 \( 1 + 3.20iT - 8T^{2} \)
5 \( 1 + 11.9T + 125T^{2} \)
11 \( 1 + 60.3iT - 1.33e3T^{2} \)
13 \( 1 + 3.46iT - 2.19e3T^{2} \)
17 \( 1 + 32.4T + 4.91e3T^{2} \)
19 \( 1 + 142. iT - 6.85e3T^{2} \)
23 \( 1 + 89.8iT - 1.21e4T^{2} \)
29 \( 1 - 247. iT - 2.43e4T^{2} \)
31 \( 1 - 207. iT - 2.97e4T^{2} \)
37 \( 1 - 98.3T + 5.06e4T^{2} \)
41 \( 1 + 150.T + 6.89e4T^{2} \)
43 \( 1 - 59.4T + 7.95e4T^{2} \)
47 \( 1 + 232.T + 1.03e5T^{2} \)
53 \( 1 + 292. iT - 1.48e5T^{2} \)
59 \( 1 - 465.T + 2.05e5T^{2} \)
61 \( 1 + 13.1iT - 2.26e5T^{2} \)
67 \( 1 - 481.T + 3.00e5T^{2} \)
71 \( 1 + 550. iT - 3.57e5T^{2} \)
73 \( 1 + 372. iT - 3.89e5T^{2} \)
79 \( 1 - 879.T + 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 21.2T + 7.04e5T^{2} \)
97 \( 1 - 612. iT - 9.12e5T^{2} \)
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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

−11.67153778968934932255727002020, −11.11809650518521882154936214863, −10.53301212947781110347894804253, −9.117019384484547315727598196488, −8.391167214907803738212828274892, −6.73031569512668293329857805061, −4.91164681129878596715290607700, −3.66646044634294304386046854137, −2.88787722312911889750921243233, −0.40144609638449387156449934666, 2.08100094432720455692244011374, 4.14345117405642074742479104618, 5.75755678810671030168048145061, 6.88037650391134527874300391351, 7.71361705229995557849100955118, 8.152459474168257227351602537244, 9.697740146807216441238979515557, 11.46352321315988746676937331206, 12.02572133179818000495830569786, 13.09966381744388585105316518965

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