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Properties

Label 2-147-7.3-c6-0-21
Degree $2$
Conductor $147$
Sign $-0.553 + 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.23 − 10.8i)2-s + (−13.5 − 7.79i)3-s + (−45.8 + 79.3i)4-s + (175. − 101. i)5-s + 194. i·6-s + 345.·8-s + (121.5 + 210. i)9-s + (−2.18e3 − 1.26e3i)10-s + (−437. + 758. i)11-s + (1.23e3 − 714. i)12-s + 275. i·13-s − 3.15e3·15-s + (780. + 1.35e3i)16-s + (3.79e3 + 2.19e3i)17-s + (1.51e3 − 2.62e3i)18-s + (1.16e4 − 6.75e3i)19-s + ⋯
L(s)  = 1  + (−0.779 − 1.35i)2-s + (−0.5 − 0.288i)3-s + (−0.716 + 1.24i)4-s + (1.40 − 0.809i)5-s + 0.900i·6-s + 0.674·8-s + (0.166 + 0.288i)9-s + (−2.18 − 1.26i)10-s + (−0.328 + 0.569i)11-s + (0.716 − 0.413i)12-s + 0.125i·13-s − 0.935·15-s + (0.190 + 0.329i)16-s + (0.773 + 0.446i)17-s + (0.259 − 0.450i)18-s + (1.70 − 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.457450446\)
\(L(\frac12)\) \(\approx\) \(1.457450446\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (6.23 + 10.8i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-175. + 101. i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (437. - 758. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 275. iT - 4.82e6T^{2} \)
17 \( 1 + (-3.79e3 - 2.19e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.16e4 + 6.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-6.36e3 - 1.10e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 6.26e3T + 5.94e8T^{2} \)
31 \( 1 + (-1.76e4 - 1.02e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (7.86e3 + 1.36e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 6.99e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.13e5T + 6.32e9T^{2} \)
47 \( 1 + (-4.01e4 + 2.31e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (6.40e4 - 1.10e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (2.42e5 + 1.40e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (8.47e4 - 4.89e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (8.72e4 - 1.51e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.45e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.04e5 - 6.03e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (3.81e5 + 6.61e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 8.59e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.55e5 + 8.99e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 3.40e5iT - 8.32e11T^{2} \)
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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

−11.55003037969726956173501686913, −10.43348353312411349225410250500, −9.644853843749638283978502121470, −9.047033764386503498020529295029, −7.58878289349634906438300195922, −5.93501985582168910459762295712, −4.89694826542435441375125652100, −2.89248258070465242086191437765, −1.61115997845326732574353243943, −0.897265493756241526355295301837, 0.923014931527965953673955345324, 2.95810999235717902687315655922, 5.34247919943477163163211652340, 5.88795683465937307749379694461, 6.85094126762745032787062740352, 7.903613173731166387365184466726, 9.299185413106726030055072785678, 9.942487999970711415150876472285, 10.78348264755187789265609502645, 12.25026510112081264466126899433

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