Properties

Label 2-147-7.4-c3-0-14
Degree $2$
Conductor $147$
Sign $0.0725 + 0.997i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 + 0.358i)2-s + (1.5 − 2.59i)3-s + (3.91 − 6.77i)4-s + (−0.0502 − 0.0870i)5-s + 1.24·6-s + 6.55·8-s + (−4.5 − 7.79i)9-s + (0.0208 − 0.0360i)10-s + (21.9 − 38.0i)11-s + (−11.7 − 20.3i)12-s + 16.6·13-s − 0.301·15-s + (−29.9 − 51.8i)16-s + (−60.8 + 105. i)17-s + (1.86 − 3.22i)18-s + (−63.5 − 110. i)19-s + ⋯
L(s)  = 1  + (0.0732 + 0.126i)2-s + (0.288 − 0.499i)3-s + (0.489 − 0.847i)4-s + (−0.00449 − 0.00778i)5-s + 0.0845·6-s + 0.289·8-s + (−0.166 − 0.288i)9-s + (0.000658 − 0.00114i)10-s + (0.602 − 1.04i)11-s + (−0.282 − 0.489i)12-s + 0.355·13-s − 0.00519·15-s + (−0.468 − 0.810i)16-s + (−0.867 + 1.50i)17-s + (0.0244 − 0.0422i)18-s + (−0.767 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.49263 - 1.38798i\)
\(L(\frac12)\) \(\approx\) \(1.49263 - 1.38798i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (-0.207 - 0.358i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (0.0502 + 0.0870i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-21.9 + 38.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 16.6T + 2.19e3T^{2} \)
17 \( 1 + (60.8 - 105. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (63.5 + 110. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (26.7 + 46.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 235.T + 2.43e4T^{2} \)
31 \( 1 + (9.35 - 16.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-95.9 - 166. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 319.T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 + (-200. - 347. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (321. - 556. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (5.80 - 10.0i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (6.12 + 10.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (334. - 579. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 822.T + 3.57e5T^{2} \)
73 \( 1 + (-257. + 446. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-402. - 697. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 394.T + 5.71e5T^{2} \)
89 \( 1 + (-336. - 583. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−12.41428982305772586234322291312, −11.15647887838327756243597650522, −10.57415352025727595387036645031, −9.061544050125883971646793509390, −8.224517840771322854769259508601, −6.53866062484784428824744584869, −6.23352535475125014326339958920, −4.44075063925451579994959583955, −2.56805111653701323577477677954, −0.988127304698986026293864756835, 2.15369609537345591673456908489, 3.60180789755293982454707128600, 4.69919174740475743883064768881, 6.53594560152330192005493333844, 7.55855865675264822404565869907, 8.701910731309028746738699624463, 9.722573389511194087519854139991, 10.91440993940435912139998259454, 11.84025510933883378626906178289, 12.69883938367124801632328833806

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