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Properties

Label 2-147-147.104-c1-0-10
Degree $2$
Conductor $147$
Sign $-0.936 - 0.350i$
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  + (−1.78 − 1.42i)2-s + (0.0426 + 1.73i)3-s + (0.710 + 3.11i)4-s + (−1.09 − 0.527i)5-s + (2.38 − 3.14i)6-s + (−2.21 − 1.44i)7-s + (1.17 − 2.44i)8-s + (−2.99 + 0.147i)9-s + (1.20 + 2.49i)10-s + (−3.63 − 2.90i)11-s + (−5.35 + 1.36i)12-s + (−2.21 − 1.76i)13-s + (1.90 + 5.71i)14-s + (0.866 − 1.91i)15-s + (0.175 − 0.0846i)16-s + (−0.896 + 3.92i)17-s + ⋯
L(s)  = 1  + (−1.25 − 1.00i)2-s + (0.0246 + 0.999i)3-s + (0.355 + 1.55i)4-s + (−0.489 − 0.235i)5-s + (0.973 − 1.28i)6-s + (−0.838 − 0.544i)7-s + (0.416 − 0.865i)8-s + (−0.998 + 0.0492i)9-s + (0.380 + 0.789i)10-s + (−1.09 − 0.874i)11-s + (−1.54 + 0.393i)12-s + (−0.613 − 0.489i)13-s + (0.508 + 1.52i)14-s + (0.223 − 0.495i)15-s + (0.0439 − 0.0211i)16-s + (−0.217 + 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00671970 + 0.0370867i\)
\(L(\frac12)\) \(\approx\) \(0.00671970 + 0.0370867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0426 - 1.73i)T \)
7 \( 1 + (2.21 + 1.44i)T \)
good2 \( 1 + (1.78 + 1.42i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.09 + 0.527i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (3.63 + 2.90i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.21 + 1.76i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.896 - 3.92i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 5.20iT - 19T^{2} \)
23 \( 1 + (3.01 - 0.687i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-5.02 - 1.14i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 + (-0.330 + 1.44i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.32 - 2.56i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (9.88 - 4.76i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.464 - 0.582i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.20 + 0.503i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-6.73 + 3.24i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-8.54 - 1.95i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 6.55T + 67T^{2} \)
71 \( 1 + (2.32 - 0.530i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.07 + 1.65i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (-1.42 - 1.78i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (11.4 + 14.3i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 2.44iT - 97T^{2} \)
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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

−12.12438190033071608061914286768, −11.09512219283825954549171322257, −10.17473159530002862961217120678, −9.900003047960038557758670718369, −8.433888022222303161929183872851, −7.933137874878345103524316429727, −5.81465393478966479965488739704, −3.97414307885573480194772745801, −2.85844333789247989674899569092, −0.05138174201644863684273382241, 2.56953067614981612096083772848, 5.33435677021959494749088062488, 6.81118401469117237968547412841, 7.15667972552017242309083064301, 8.243325479399784142897513811815, 9.229713834716498083745388829432, 10.18347808547281741850530881540, 11.64325354883237032547564752030, 12.58031636269923118605624293568, 13.65577452736810317581382554488

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