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Properties

Label 2-147-3.2-c2-0-21
Degree $2$
Conductor $147$
Sign $-0.274 - 0.961i$
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  − 3.50i·2-s + (−0.822 − 2.88i)3-s − 8.29·4-s − 1.24i·5-s + (−10.1 + 2.88i)6-s + 15.0i·8-s + (−7.64 + 4.74i)9-s − 4.35·10-s − 7.01i·11-s + (6.82 + 23.9i)12-s + 11.6·13-s + (−3.58 + 1.02i)15-s + 19.5·16-s − 4.52i·17-s + (16.6 + 26.8i)18-s − 16.2·19-s + ⋯
L(s)  = 1  − 1.75i·2-s + (−0.274 − 0.961i)3-s − 2.07·4-s − 0.248i·5-s + (−1.68 + 0.480i)6-s + 1.88i·8-s + (−0.849 + 0.527i)9-s − 0.435·10-s − 0.637i·11-s + (0.568 + 1.99i)12-s + 0.895·13-s + (−0.238 + 0.0681i)15-s + 1.22·16-s − 0.266i·17-s + (0.924 + 1.48i)18-s − 0.854·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ -0.274 - 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.539850 + 0.715364i\)
\(L(\frac12)\) \(\approx\) \(0.539850 + 0.715364i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.822 + 2.88i)T \)
7 \( 1 \)
good2 \( 1 + 3.50iT - 4T^{2} \)
5 \( 1 + 1.24iT - 25T^{2} \)
11 \( 1 + 7.01iT - 121T^{2} \)
13 \( 1 - 11.6T + 169T^{2} \)
17 \( 1 + 4.52iT - 289T^{2} \)
19 \( 1 + 16.2T + 361T^{2} \)
23 \( 1 + 25.5iT - 529T^{2} \)
29 \( 1 - 9.49iT - 841T^{2} \)
31 \( 1 + 28.7T + 961T^{2} \)
37 \( 1 + 33.0T + 1.36e3T^{2} \)
41 \( 1 + 67.1iT - 1.68e3T^{2} \)
43 \( 1 + 24.1T + 1.84e3T^{2} \)
47 \( 1 - 33.0iT - 2.20e3T^{2} \)
53 \( 1 + 15.1iT - 2.80e3T^{2} \)
59 \( 1 + 92.3iT - 3.48e3T^{2} \)
61 \( 1 - 57.5T + 3.72e3T^{2} \)
67 \( 1 - 15.1T + 4.48e3T^{2} \)
71 \( 1 + 70.5iT - 5.04e3T^{2} \)
73 \( 1 - 76.7T + 5.32e3T^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 + 74.2iT - 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−12.20485721454303434314383722201, −11.04996470742109711438479942118, −10.66160067398528123564502414375, −9.015102905788156056179047243540, −8.351045360486225723424964728757, −6.58904996185403290438421037282, −5.10553957964145859483028829590, −3.50288480088276159845654345278, −2.06224856842865433615623294812, −0.61836142257722640622445411175, 3.80252213255153466524122611574, 4.96493586571976555451175980518, 5.99563410038785757901848089072, 6.95948992208732989080967161971, 8.286910992971234382946137325075, 9.127646934687828337516767281944, 10.19472578880054647455273852929, 11.34752956887589062561448680833, 12.89480226269174544753570156025, 13.96402845487039010114483596574

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