Properties

Label 2-147-49.32-c1-0-3
Degree $2$
Conductor $147$
Sign $0.988 + 0.152i$
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  + (−0.0443 − 0.591i)2-s + (0.955 + 0.294i)3-s + (1.62 − 0.245i)4-s + (−1.97 + 1.82i)5-s + (0.131 − 0.578i)6-s + (2.50 + 0.864i)7-s + (−0.481 − 2.10i)8-s + (0.826 + 0.563i)9-s + (1.16 + 1.08i)10-s + (0.280 − 0.191i)11-s + (1.62 + 0.245i)12-s + (−5.54 − 2.67i)13-s + (0.400 − 1.51i)14-s + (−2.42 + 1.16i)15-s + (1.92 − 0.593i)16-s + (0.335 + 0.855i)17-s + ⋯
L(s)  = 1  + (−0.0313 − 0.418i)2-s + (0.551 + 0.170i)3-s + (0.814 − 0.122i)4-s + (−0.881 + 0.817i)5-s + (0.0538 − 0.236i)6-s + (0.945 + 0.326i)7-s + (−0.170 − 0.745i)8-s + (0.275 + 0.187i)9-s + (0.369 + 0.342i)10-s + (0.0846 − 0.0577i)11-s + (0.470 + 0.0709i)12-s + (−1.53 − 0.740i)13-s + (0.107 − 0.405i)14-s + (−0.625 + 0.301i)15-s + (0.480 − 0.148i)16-s + (0.0814 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37767 - 0.105680i\)
\(L(\frac12)\) \(\approx\) \(1.37767 - 0.105680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (-2.50 - 0.864i)T \)
good2 \( 1 + (0.0443 + 0.591i)T + (-1.97 + 0.298i)T^{2} \)
5 \( 1 + (1.97 - 1.82i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-0.280 + 0.191i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (5.54 + 2.67i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.335 - 0.855i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (1.29 + 2.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.07 + 5.27i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (5.27 - 6.61i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.94 - 5.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.78 + 0.871i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (2.03 + 8.92i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.40 - 6.17i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.137 - 1.84i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (-5.42 + 0.817i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-6.66 - 6.18i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-7.71 - 1.16i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (-4.27 + 7.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.39 + 6.76i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.680 + 9.07i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (0.201 + 0.348i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.89 - 2.83i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-11.9 - 8.15i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 - 5.14T + 97T^{2} \)
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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

−12.69670696565660158412447313169, −11.93906019749674674562292934235, −10.90508152135742606306621146384, −10.38772539192186931961417830435, −8.860130488792824881666585929170, −7.59929376177045092144368972307, −6.98959310824780535191522443034, −5.09873915060030601122088649505, −3.43473948442251813895828667553, −2.30158255485333767213250244529, 2.02762445241098382411900454532, 4.01086281883742933389022043043, 5.27692378982672863266891673473, 7.12119624642401196028106498977, 7.70809888698506877047812272601, 8.561860507717577973447986719789, 9.890273195791181422993687444245, 11.55660948225443357481952732880, 11.80105532555725009285338823917, 13.04132984935268251068048337897

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