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Properties

Label 2-147-21.20-c5-0-36
Degree $2$
Conductor $147$
Sign $0.849 - 0.528i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52i·2-s + (15.4 − 2.09i)3-s + 29.6·4-s + 32.5·5-s + (3.19 + 23.4i)6-s + 93.8i·8-s + (234. − 64.8i)9-s + 49.4i·10-s + 321. i·11-s + (458. − 62.3i)12-s + 609. i·13-s + (502. − 68.2i)15-s + 807.·16-s + 31.0·17-s + (98.6 + 356. i)18-s − 1.31e3i·19-s + ⋯
L(s)  = 1  + 0.268i·2-s + (0.990 − 0.134i)3-s + 0.927·4-s + 0.581·5-s + (0.0362 + 0.266i)6-s + 0.518i·8-s + (0.963 − 0.266i)9-s + 0.156i·10-s + 0.801i·11-s + (0.919 − 0.124i)12-s + 0.999i·13-s + (0.576 − 0.0783i)15-s + 0.788·16-s + 0.0260·17-s + (0.0717 + 0.259i)18-s − 0.836i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.849 - 0.528i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.849 - 0.528i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.874779445\)
\(L(\frac12)\) \(\approx\) \(3.874779445\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.4 + 2.09i)T \)
7 \( 1 \)
good2 \( 1 - 1.52iT - 32T^{2} \)
5 \( 1 - 32.5T + 3.12e3T^{2} \)
11 \( 1 - 321. iT - 1.61e5T^{2} \)
13 \( 1 - 609. iT - 3.71e5T^{2} \)
17 \( 1 - 31.0T + 1.41e6T^{2} \)
19 \( 1 + 1.31e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.06e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.10e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.85e3iT - 2.86e7T^{2} \)
37 \( 1 + 9.76e3T + 6.93e7T^{2} \)
41 \( 1 - 1.96e4T + 1.15e8T^{2} \)
43 \( 1 + 9.70e3T + 1.47e8T^{2} \)
47 \( 1 - 1.82e4T + 2.29e8T^{2} \)
53 \( 1 - 3.31e4iT - 4.18e8T^{2} \)
59 \( 1 + 6.67e3T + 7.14e8T^{2} \)
61 \( 1 + 3.50e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.93e4T + 1.35e9T^{2} \)
71 \( 1 + 2.14e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.29e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.14e3T + 3.07e9T^{2} \)
83 \( 1 + 6.50e4T + 3.93e9T^{2} \)
89 \( 1 + 1.14e4T + 5.58e9T^{2} \)
97 \( 1 + 1.32e5iT - 8.58e9T^{2} \)
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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

−12.26906593979157437869038361385, −11.28571588262926688175233755514, −9.910637236705519555089520062586, −9.222009899519869403453372449442, −7.79365422334460297338749625324, −7.06635868929621498213723352945, −5.92740633081090591918728498288, −4.19519749321872292204485018594, −2.53967969855416350599848121876, −1.70803182966496003079264402872, 1.33843369179580353741416245103, 2.62655747651169454057897287266, 3.58717443298158413816387118466, 5.51064656325702127713093685956, 6.76589422043622192143807060292, 7.934576509036438273592298885924, 8.912294713447566263927156368904, 10.26880727458733579990251144654, 10.66734544048630131455053523263, 12.22618419401422230698568979553

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