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Properties

Label 2-147-21.20-c5-0-58
Degree $2$
Conductor $147$
Sign $-0.573 - 0.818i$
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  − 8.31i·2-s + (14.0 − 6.83i)3-s − 37.1·4-s − 28.6·5-s + (−56.8 − 116. i)6-s + 42.3i·8-s + (149. − 191. i)9-s + 238. i·10-s − 596. i·11-s + (−519. + 253. i)12-s − 602. i·13-s + (−401. + 196. i)15-s − 834.·16-s + 154.·17-s + (−1.59e3 − 1.24e3i)18-s + 2.86e3i·19-s + ⋯
L(s)  = 1  − 1.46i·2-s + (0.898 − 0.438i)3-s − 1.15·4-s − 0.513·5-s + (−0.644 − 1.32i)6-s + 0.234i·8-s + (0.615 − 0.788i)9-s + 0.754i·10-s − 1.48i·11-s + (−1.04 + 0.508i)12-s − 0.988i·13-s + (−0.461 + 0.225i)15-s − 0.815·16-s + 0.129·17-s + (−1.15 − 0.904i)18-s + 1.82i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.818i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.573 - 0.818i$
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: \((2,\ 147,\ (\ :5/2),\ -0.573 - 0.818i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.737966014\)
\(L(\frac12)\) \(\approx\) \(1.737966014\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.0 + 6.83i)T \)
7 \( 1 \)
good2 \( 1 + 8.31iT - 32T^{2} \)
5 \( 1 + 28.6T + 3.12e3T^{2} \)
11 \( 1 + 596. iT - 1.61e5T^{2} \)
13 \( 1 + 602. iT - 3.71e5T^{2} \)
17 \( 1 - 154.T + 1.41e6T^{2} \)
19 \( 1 - 2.86e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.78e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.85e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.22e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.04e4T + 6.93e7T^{2} \)
41 \( 1 + 6.92e3T + 1.15e8T^{2} \)
43 \( 1 - 3.80e3T + 1.47e8T^{2} \)
47 \( 1 - 1.57e4T + 2.29e8T^{2} \)
53 \( 1 + 4.78e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.34e4T + 7.14e8T^{2} \)
61 \( 1 + 4.51e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.95e4T + 1.35e9T^{2} \)
71 \( 1 + 3.59e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.67e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.05e3T + 3.07e9T^{2} \)
83 \( 1 - 3.62e4T + 3.93e9T^{2} \)
89 \( 1 - 7.63e4T + 5.58e9T^{2} \)
97 \( 1 + 5.19e4iT - 8.58e9T^{2} \)
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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

−11.65188339382437985972401278205, −10.52516429873967837019093553481, −9.619223716011959293621594174023, −8.435075542801821340883782424189, −7.64756277310057608220927351930, −5.86379253329199021023252385137, −3.72017855290844555337991239136, −3.30340951809081792423138486185, −1.78533236293613374808074709750, −0.51574989127411784404867736323, 2.31303060950545163854569927782, 4.24052745381365302478500142369, 4.95280336983184359835980275803, 6.87532680747972025371900572654, 7.30320075550936058803405252217, 8.579472992510139660807483140883, 9.194827982035599465556132803089, 10.48437881883470860905538944222, 11.93373084893535618200888348762, 13.22815590550863334210579554305

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