Properties

Label 2-147-147.23-c2-0-15
Degree $2$
Conductor $147$
Sign $0.930 + 0.367i$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.86 + 0.884i)3-s + (−3.95 − 0.596i)4-s + (6.98 − 0.479i)7-s + (7.43 − 5.06i)9-s + (11.8 − 1.78i)12-s + (15.2 − 7.34i)13-s + (15.2 + 4.71i)16-s + (12.8 − 22.2i)19-s + (−19.5 + 7.54i)21-s + (1.86 + 24.9i)25-s + (−16.8 + 21.1i)27-s + (−27.9 − 2.26i)28-s + (−19.6 − 34.0i)31-s + (−32.4 + 15.6i)36-s + (54.3 − 8.19i)37-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)3-s + (−0.988 − 0.149i)4-s + (0.997 − 0.0684i)7-s + (0.826 − 0.563i)9-s + (0.988 − 0.149i)12-s + (1.17 − 0.565i)13-s + (0.955 + 0.294i)16-s + (0.676 − 1.17i)19-s + (−0.933 + 0.359i)21-s + (0.0747 + 0.997i)25-s + (−0.623 + 0.781i)27-s + (−0.996 − 0.0809i)28-s + (−0.633 − 1.09i)31-s + (−0.900 + 0.433i)36-s + (1.46 − 0.221i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 0.930 + 0.367i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.947423 - 0.180154i\)
\(L(\frac12)\) \(\approx\) \(0.947423 - 0.180154i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.86 - 0.884i)T \)
7 \( 1 + (-6.98 + 0.479i)T \)
good2 \( 1 + (3.95 + 0.596i)T^{2} \)
5 \( 1 + (-1.86 - 24.9i)T^{2} \)
11 \( 1 + (-44.2 - 112. i)T^{2} \)
13 \( 1 + (-15.2 + 7.34i)T + (105. - 132. i)T^{2} \)
17 \( 1 + (211. + 196. i)T^{2} \)
19 \( 1 + (-12.8 + 22.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (387. - 359. i)T^{2} \)
29 \( 1 + (187. - 819. i)T^{2} \)
31 \( 1 + (19.6 + 34.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-54.3 + 8.19i)T + (1.30e3 - 403. i)T^{2} \)
41 \( 1 + (1.51e3 + 729. i)T^{2} \)
43 \( 1 + (6.37 + 27.9i)T + (-1.66e3 + 802. i)T^{2} \)
47 \( 1 + (2.18e3 + 329. i)T^{2} \)
53 \( 1 + (-2.68e3 - 827. i)T^{2} \)
59 \( 1 + (-260. + 3.47e3i)T^{2} \)
61 \( 1 + (-29.3 + 4.42i)T + (3.55e3 - 1.09e3i)T^{2} \)
67 \( 1 + (-63.5 - 110. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (10.8 + 145. i)T + (-5.26e3 + 794. i)T^{2} \)
79 \( 1 + (65.0 - 112. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-2.89e3 + 7.37e3i)T^{2} \)
97 \( 1 - 82.3T + 9.40e3T^{2} \)
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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

−12.88748152254135287512156219500, −11.48002607279440289224693611915, −10.92345431114003520310755381459, −9.728824357991175481380190773459, −8.734339004418985182447802246825, −7.48357522009267968394871934654, −5.84155855414043197860119027099, −5.01530504003106344380935810094, −3.89220654229761673529158218249, −0.943915196812159398961191086541, 1.28901710124743020512965656500, 4.03011701090035638100165985222, 5.09593626189726577241003363464, 6.16944002732434782506812015106, 7.71071564154297430328604380231, 8.593105815150077214576609262272, 9.920455867145430392949127832438, 10.99839486185027888358039567880, 11.88737329503643406131987599920, 12.80067554742219605554606459078

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