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Properties

Label 2-147-7.4-c5-0-1
Degree $2$
Conductor $147$
Sign $-0.386 + 0.922i$
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  + (4.22 + 7.31i)2-s + (−4.5 + 7.79i)3-s + (−19.6 + 34.0i)4-s + (18 + 31.1i)5-s − 76.0·6-s − 61.9·8-s + (−40.5 − 70.1i)9-s + (−152. + 263. i)10-s + (−147. + 255. i)11-s + (−177. − 306. i)12-s − 1.14e3·13-s − 324·15-s + (367. + 636. i)16-s + (−516. + 894. i)17-s + (342. − 592. i)18-s + (−1.05e3 − 1.82e3i)19-s + ⋯
L(s)  = 1  + (0.746 + 1.29i)2-s + (−0.288 + 0.499i)3-s + (−0.614 + 1.06i)4-s + (0.321 + 0.557i)5-s − 0.862·6-s − 0.342·8-s + (−0.166 − 0.288i)9-s + (−0.480 + 0.832i)10-s + (−0.368 + 0.637i)11-s + (−0.354 − 0.614i)12-s − 1.88·13-s − 0.371·15-s + (0.359 + 0.621i)16-s + (−0.433 + 0.750i)17-s + (0.248 − 0.431i)18-s + (−0.669 − 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.520598412\)
\(L(\frac12)\) \(\approx\) \(1.520598412\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-4.22 - 7.31i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-18 - 31.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (147. - 255. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.14e3T + 3.71e5T^{2} \)
17 \( 1 + (516. - 894. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.05e3 + 1.82e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-320. - 555. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.63e3T + 2.05e7T^{2} \)
31 \( 1 + (-483. + 837. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-886. - 1.53e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.19e4T + 1.15e8T^{2} \)
43 \( 1 + 1.98e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.39e4 - 2.42e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-3.55e3 + 6.16e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.04e4 + 1.80e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.19e4 - 2.06e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.73e4 - 3.00e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.84e4T + 1.80e9T^{2} \)
73 \( 1 + (-7.64e3 + 1.32e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.65e4 - 6.32e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 3.03e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.80e4 - 3.12e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.53e5T + 8.58e9T^{2} \)
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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

−13.06621456514232668367918597356, −12.11316948282613906186522055902, −10.63009199973573488105341724493, −9.872105933168760621161376766547, −8.377655474402587694959674414487, −7.08836076275241121556252795694, −6.46792368311945018344141513372, −5.08375222837157653917740167971, −4.45861276921992137466740446258, −2.57759338443716562144763363260, 0.38972979076934287246565437279, 1.84892524201049896795081606936, 2.94701482287230663160190025542, 4.64426629175017178932351141628, 5.40684809445614678000790239515, 7.01453773230525388392603111734, 8.404747180072119851763403820929, 9.844235011302412812932664281277, 10.59536207767189878388814371645, 11.88959269315441161741992438957

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