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Properties

Label 2-147-21.2-c2-0-18
Degree $2$
Conductor $147$
Sign $0.424 + 0.905i$
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  + (0.679 + 0.392i)2-s + (2.91 − 0.709i)3-s + (−1.69 − 2.93i)4-s + (−4.02 − 2.32i)5-s + (2.25 + 0.661i)6-s − 5.79i·8-s + (7.99 − 4.13i)9-s + (−1.82 − 3.15i)10-s + (16.8 − 9.71i)11-s + (−7.01 − 7.34i)12-s − 8.75·13-s + (−13.3 − 3.92i)15-s + (−4.5 + 7.79i)16-s + (7.09 − 4.09i)17-s + (7.04 + 0.326i)18-s + (−12.4 + 21.5i)19-s + ⋯
L(s)  = 1  + (0.339 + 0.196i)2-s + (0.971 − 0.236i)3-s + (−0.423 − 0.732i)4-s + (−0.805 − 0.464i)5-s + (0.376 + 0.110i)6-s − 0.723i·8-s + (0.888 − 0.459i)9-s + (−0.182 − 0.315i)10-s + (1.52 − 0.882i)11-s + (−0.584 − 0.612i)12-s − 0.673·13-s + (−0.892 − 0.261i)15-s + (−0.281 + 0.487i)16-s + (0.417 − 0.240i)17-s + (0.391 + 0.0181i)18-s + (−0.654 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.424 + 0.905i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 0.424 + 0.905i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59957 - 1.01711i\)
\(L(\frac12)\) \(\approx\) \(1.59957 - 1.01711i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.91 + 0.709i)T \)
7 \( 1 \)
good2 \( 1 + (-0.679 - 0.392i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (4.02 + 2.32i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-16.8 + 9.71i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 8.75T + 169T^{2} \)
17 \( 1 + (-7.09 + 4.09i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (12.4 - 21.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.3 - 8.29i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 21.2iT - 841T^{2} \)
31 \( 1 + (-16.6 - 28.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (6.42 - 11.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 33.0iT - 1.68e3T^{2} \)
43 \( 1 - 46.6T + 1.84e3T^{2} \)
47 \( 1 + (-59.4 - 34.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (29.5 - 17.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (29.5 - 17.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.57 - 9.65i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.45 - 9.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 21.2iT - 5.04e3T^{2} \)
73 \( 1 + (41.6 + 72.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (14.7 - 25.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 75.2iT - 6.88e3T^{2} \)
89 \( 1 + (78.4 + 45.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 26.6T + 9.40e3T^{2} \)
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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

−12.68463104521085543974163571125, −11.98297147600087204434442986923, −10.45612968543936156184964167728, −9.244996919488822400665807195419, −8.625540161723376473886494153376, −7.34835222901676720085919744980, −6.11705775749041412077459493779, −4.50516021701054005711784718709, −3.54910075952624404423998635996, −1.17825407882339451627029209478, 2.55504700623319599867657441334, 3.87362659469543081640483412897, 4.53482876608459682600880540604, 6.94843700072384454799190568085, 7.74930823615450827432342491320, 8.868871467950146851600201930968, 9.689249067250598617412292192992, 11.20408546054237885741315775445, 12.13888767467117165152211658362, 12.97357586285228648801109303324

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