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Properties

Label 2-147-21.20-c5-0-0
Degree $2$
Conductor $147$
Sign $0.332 + 0.943i$
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  + 4.29i·2-s + (−1.28 + 15.5i)3-s + 13.5·4-s − 75.7·5-s + (−66.6 − 5.52i)6-s + 195. i·8-s + (−239. − 40.0i)9-s − 324. i·10-s + 683. i·11-s + (−17.5 + 211. i)12-s − 904. i·13-s + (97.5 − 1.17e3i)15-s − 404.·16-s − 831.·17-s + (171. − 1.02e3i)18-s − 46.6i·19-s + ⋯
L(s)  = 1  + 0.758i·2-s + (−0.0825 + 0.996i)3-s + 0.424·4-s − 1.35·5-s + (−0.755 − 0.0626i)6-s + 1.08i·8-s + (−0.986 − 0.164i)9-s − 1.02i·10-s + 1.70i·11-s + (−0.0350 + 0.423i)12-s − 1.48i·13-s + (0.111 − 1.34i)15-s − 0.394·16-s − 0.697·17-s + (0.124 − 0.748i)18-s − 0.0296i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.332 + 0.943i$
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: \((2,\ 147,\ (\ :5/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1869935926\)
\(L(\frac12)\) \(\approx\) \(0.1869935926\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 15.5i)T \)
7 \( 1 \)
good2 \( 1 - 4.29iT - 32T^{2} \)
5 \( 1 + 75.7T + 3.12e3T^{2} \)
11 \( 1 - 683. iT - 1.61e5T^{2} \)
13 \( 1 + 904. iT - 3.71e5T^{2} \)
17 \( 1 + 831.T + 1.41e6T^{2} \)
19 \( 1 + 46.6iT - 2.47e6T^{2} \)
23 \( 1 + 3.22e3iT - 6.43e6T^{2} \)
29 \( 1 - 644. iT - 2.05e7T^{2} \)
31 \( 1 - 818. iT - 2.86e7T^{2} \)
37 \( 1 - 1.25e4T + 6.93e7T^{2} \)
41 \( 1 + 3.58e3T + 1.15e8T^{2} \)
43 \( 1 + 1.27e4T + 1.47e8T^{2} \)
47 \( 1 - 5.08e3T + 2.29e8T^{2} \)
53 \( 1 + 2.92e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.42e4T + 7.14e8T^{2} \)
61 \( 1 - 1.02e3iT - 8.44e8T^{2} \)
67 \( 1 + 6.56e3T + 1.35e9T^{2} \)
71 \( 1 + 2.16e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.61e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.45e4T + 3.07e9T^{2} \)
83 \( 1 + 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + 1.20e4T + 5.58e9T^{2} \)
97 \( 1 - 1.31e5iT - 8.58e9T^{2} \)
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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

−12.74869750486089513589539142314, −11.80811555343756893292329577271, −10.91834697365677272769439191879, −10.01063597528495768105971247657, −8.489989433466123219265975043360, −7.73880294262258732320906075180, −6.65523804492098805017182470238, −5.16174548060585069648976691342, −4.22648151257293643908636613803, −2.72255658329901886893197552849, 0.06418457287978335987803504623, 1.36846938214234819755346834524, 2.89831289878840979204456459288, 4.00982292116556005436739873174, 6.08586205381863178758709018230, 7.07544834046876899528250199749, 8.036814070211465320338010909702, 9.129807670085293898887900189984, 10.98701510542629171588844095988, 11.51036109572201462064840964177

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