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Properties

Label 2-147-7.4-c3-0-1
Degree $2$
Conductor $147$
Sign $-0.386 + 0.922i$
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  + (2.63 + 4.56i)2-s + (−1.5 + 2.59i)3-s + (−9.91 + 17.1i)4-s + (−5.27 − 9.13i)5-s − 15.8·6-s − 62.3·8-s + (−4.5 − 7.79i)9-s + (27.8 − 48.1i)10-s + (−17.3 + 30.0i)11-s + (−29.7 − 51.5i)12-s − 37.2·13-s + 31.6·15-s + (−85.2 − 147. i)16-s + (5.27 − 9.13i)17-s + (23.7 − 41.1i)18-s + (29.2 + 50.7i)19-s + ⋯
L(s)  = 1  + (0.932 + 1.61i)2-s + (−0.288 + 0.499i)3-s + (−1.23 + 2.14i)4-s + (−0.471 − 0.817i)5-s − 1.07·6-s − 2.75·8-s + (−0.166 − 0.288i)9-s + (0.879 − 1.52i)10-s + (−0.476 + 0.824i)11-s + (−0.715 − 1.23i)12-s − 0.795·13-s + 0.544·15-s + (−1.33 − 2.30i)16-s + (0.0752 − 0.130i)17-s + (0.310 − 0.538i)18-s + (0.353 + 0.612i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 147,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.664486 - 0.998954i\)
\(L(\frac12)\) \(\approx\) \(0.664486 - 0.998954i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.63 - 4.56i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (5.27 + 9.13i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (17.3 - 30.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 37.2T + 2.19e3T^{2} \)
17 \( 1 + (-5.27 + 9.13i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-29.2 - 50.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-62.6 - 108. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 35.4T + 2.43e4T^{2} \)
31 \( 1 + (145. - 252. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-129. - 225. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 338.T + 6.89e4T^{2} \)
43 \( 1 - 6.80T + 7.95e4T^{2} \)
47 \( 1 + (125. + 217. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-268. + 464. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-17.9 + 31.0i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (28.8 + 50.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (240. - 417. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 363.T + 3.57e5T^{2} \)
73 \( 1 + (290. - 503. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-346. - 600. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + (-176. - 305. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.44e3T + 9.12e5T^{2} \)
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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

−13.37702951669618568021723945431, −12.52195642246480216051441627792, −11.81397516639466351648693653305, −9.943769409145671399123520229464, −8.760719017446982162370182245723, −7.75325424186489321260587495585, −6.82775709170678877849820223506, −5.26264740013150256294273396789, −4.87754027020575796590093327537, −3.58241608040808237628814492452, 0.43905095468108892615897437985, 2.36401543535289497793551425317, 3.37567009777173051964010573087, 4.83334181100551577078033078384, 6.02841152472127532824674950929, 7.48338794200151657963292108352, 9.177907353008261095770822983213, 10.49198908651384197133730659028, 11.08947809384677211563222141585, 11.82172499369499580240904690936

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