Properties

Label 2-147-21.5-c1-0-6
Degree $2$
Conductor $147$
Sign $0.553 + 0.832i$
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  + (1.5 − 0.866i)3-s + (−1 − 1.73i)4-s + (1.5 − 2.59i)9-s + (−3 − 1.73i)12-s + 1.73i·13-s + (−1.99 + 3.46i)16-s + (4.5 + 2.59i)19-s + (2.5 + 4.33i)25-s − 5.19i·27-s + (−7.5 + 4.33i)31-s − 6·36-s + (−0.5 + 0.866i)37-s + (1.49 + 2.59i)39-s − 5·43-s + 6.92i·48-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s + 0.480i·13-s + (−0.499 + 0.866i)16-s + (1.03 + 0.596i)19-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−1.34 + 0.777i)31-s − 36-s + (−0.0821 + 0.142i)37-s + (0.240 + 0.416i)39-s − 0.762·43-s + 0.999i·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14078 - 0.611284i\)
\(L(\frac12)\) \(\approx\) \(1.14078 - 0.611284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−13.15911253705375550662547483076, −12.06922292936953226777980827617, −10.73472382638353793359979045082, −9.560461791545669793600092713074, −8.939597098452760535590560600767, −7.66238700418792981095118759493, −6.49213545761310098579829755621, −5.11126334015221011801160834082, −3.53578764851463186617430189077, −1.59626576532413478011801504396, 2.78686444415121436799524074391, 3.93323333005488729584118466634, 5.16006046152773509895306591340, 7.20149913788669576411879604859, 8.117735917874602185360892221439, 9.026712708480955302858630794179, 9.897540922745071747189636139123, 11.17078908387755921103816931205, 12.44039791172786010664808614380, 13.31837908649765983395411742021

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