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Properties

Label 2-147-147.17-c1-0-3
Degree $2$
Conductor $147$
Sign $-0.721 - 0.692i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (−0.510 + 1.65i)3-s + (−1.97 + 0.298i)4-s + (−1.57 + 2.12i)7-s + (−2.47 − 1.68i)9-s + (0.516 − 3.42i)12-s + (−3.12 + 6.49i)13-s + (3.82 − 1.17i)16-s + (6.68 − 3.85i)19-s + (−2.70 − 3.69i)21-s + (−0.373 + 4.98i)25-s + (4.06 − 3.23i)27-s + (2.48 − 4.67i)28-s + (9.41 + 5.43i)31-s + (5.40 + 2.60i)36-s + (−9.98 − 1.50i)37-s + ⋯
L(s)  = 1  + (−0.294 + 0.955i)3-s + (−0.988 + 0.149i)4-s + (−0.596 + 0.802i)7-s + (−0.826 − 0.563i)9-s + (0.149 − 0.988i)12-s + (−0.867 + 1.80i)13-s + (0.955 − 0.294i)16-s + (1.53 − 0.885i)19-s + (−0.591 − 0.806i)21-s + (−0.0747 + 0.997i)25-s + (0.781 − 0.623i)27-s + (0.470 − 0.882i)28-s + (1.69 + 0.976i)31-s + (0.900 + 0.433i)36-s + (−1.64 − 0.247i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.721 - 0.692i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227696 + 0.566018i\)
\(L(\frac12)\) \(\approx\) \(0.227696 + 0.566018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.510 - 1.65i)T \)
7 \( 1 + (1.57 - 2.12i)T \)
good2 \( 1 + (1.97 - 0.298i)T^{2} \)
5 \( 1 + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-4.01 + 10.2i)T^{2} \)
13 \( 1 + (3.12 - 6.49i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (-6.68 + 3.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (16.8 + 15.6i)T^{2} \)
29 \( 1 + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-9.41 - 5.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.98 + 1.50i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.168 + 0.736i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-50.6 + 15.6i)T^{2} \)
59 \( 1 + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (0.987 - 6.55i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (4.68 - 8.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.23 - 0.317i)T + (72.1 + 10.8i)T^{2} \)
79 \( 1 + (0.686 + 1.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (32.5 + 82.8i)T^{2} \)
97 \( 1 + 6.40iT - 97T^{2} \)
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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

−13.61979177097188599743734876657, −12.18299177709436767563934024079, −11.63975504311197425879919982043, −10.05537043991888352892482916441, −9.320884988837582320187547926290, −8.779450170630845331412612730093, −6.95008771618662541361687792345, −5.43077655266325339542889603496, −4.57662590616347374102694159583, −3.17854264395249312638092593869, 0.65536670565407987408113987549, 3.22347665967284333046121132114, 4.99766994807132595784088316689, 6.07095848096914707351442783123, 7.52459079771552028119724135187, 8.197519662215143628530139639033, 9.815565190295303047289877388896, 10.41077761008117943681280695541, 12.04764107247069007809914264256, 12.72980490957582021793519700539

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