Properties

Label 2-147-7.4-c5-0-0
Degree $2$
Conductor $147$
Sign $-0.749 - 0.661i$
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  + (−1.69 − 2.92i)2-s + (−4.5 + 7.79i)3-s + (10.2 − 17.8i)4-s + (27.2 + 47.2i)5-s + 30.4·6-s − 177.·8-s + (−40.5 − 70.1i)9-s + (92.1 − 159. i)10-s + (−240. + 417. i)11-s + (92.5 + 160. i)12-s + 512.·13-s − 490.·15-s + (−28.8 − 49.8i)16-s + (−295. + 511. i)17-s + (−136. + 237. i)18-s + (−1.22e3 − 2.12e3i)19-s + ⋯
L(s)  = 1  + (−0.298 − 0.517i)2-s + (−0.288 + 0.499i)3-s + (0.321 − 0.556i)4-s + (0.487 + 0.844i)5-s + 0.345·6-s − 0.981·8-s + (−0.166 − 0.288i)9-s + (0.291 − 0.504i)10-s + (−0.600 + 1.03i)11-s + (0.185 + 0.321i)12-s + 0.841·13-s − 0.563·15-s + (−0.0281 − 0.0487i)16-s + (−0.247 + 0.429i)17-s + (−0.0995 + 0.172i)18-s + (−0.778 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.749 - 0.661i$
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: \((2,\ 147,\ (\ :5/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4272756531\)
\(L(\frac12)\) \(\approx\) \(0.4272756531\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.69 + 2.92i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-27.2 - 47.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (240. - 417. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 512.T + 3.71e5T^{2} \)
17 \( 1 + (295. - 511. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.22e3 + 2.12e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (887. + 1.53e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.24e3T + 2.05e7T^{2} \)
31 \( 1 + (4.88e3 - 8.45e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.98e3 - 8.63e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 3.37e3T + 1.15e8T^{2} \)
43 \( 1 + 1.82e4T + 1.47e8T^{2} \)
47 \( 1 + (660. + 1.14e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.74e4 - 3.01e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (5.79e3 - 1.00e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.57e4 + 2.71e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.37e4 - 2.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.28e4T + 1.80e9T^{2} \)
73 \( 1 + (7.96e3 - 1.38e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.35e4 + 7.54e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 9.03e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.32e4 - 1.09e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.65e4T + 8.58e9T^{2} \)
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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

−12.31277774357220334344641244209, −11.01536683377444152460517496901, −10.64372487067477541015769682322, −9.808737808358608613169708692594, −8.740011680516069725031613548174, −6.91571221903603225218304722775, −6.12375346337238107823753870727, −4.77190611721159090346440259307, −2.99475833392667575633572341474, −1.79744034849521019901075873248, 0.15158850471808925230315670640, 1.86605140714313862321342904308, 3.61012847975474235171968712627, 5.56038788476525639367237296016, 6.21472331689765563146594048875, 7.66633950519096638712929255769, 8.390587233359269993941059059446, 9.353872154033854311703440931671, 10.93190440561484011253255725951, 11.78851408930902049353863354741

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