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Properties

Label 2-147-7.2-c5-0-29
Degree $2$
Conductor $147$
Sign $-0.968 + 0.250i$
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  + (3 − 5.19i)2-s + (−4.5 − 7.79i)3-s + (−2 − 3.46i)4-s + (39 − 67.5i)5-s − 54·6-s + 168·8-s + (−40.5 + 70.1i)9-s + (−234 − 405. i)10-s + (−222 − 384. i)11-s + (−18.0 + 31.1i)12-s + 442·13-s − 702·15-s + (568 − 983. i)16-s + (−63 − 109. i)17-s + (243 + 420. i)18-s + (1.34e3 − 2.32e3i)19-s + ⋯
L(s)  = 1  + (0.530 − 0.918i)2-s + (−0.288 − 0.499i)3-s + (−0.0625 − 0.108i)4-s + (0.697 − 1.20i)5-s − 0.612·6-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (−0.739 − 1.28i)10-s + (−0.553 − 0.958i)11-s + (−0.0360 + 0.0625i)12-s + 0.725·13-s − 0.805·15-s + (0.554 − 0.960i)16-s + (−0.0528 − 0.0915i)17-s + (0.176 + 0.306i)18-s + (0.852 − 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.687999313\)
\(L(\frac12)\) \(\approx\) \(2.687999313\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3 + 5.19i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-39 + 67.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (222 + 384. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 442T + 3.71e5T^{2} \)
17 \( 1 + (63 + 109. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.34e3 + 2.32e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (2.10e3 - 3.63e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 5.44e3T + 2.05e7T^{2} \)
31 \( 1 + (-40 - 69.2i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-2.71e3 + 4.70e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 7.96e3T + 1.15e8T^{2} \)
43 \( 1 + 1.15e4T + 1.47e8T^{2} \)
47 \( 1 + (6.96e3 - 1.20e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-4.79e3 - 8.30e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.37e4 - 2.38e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.47e4 + 4.28e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.96e4 - 5.14e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.20e4T + 1.80e9T^{2} \)
73 \( 1 + (3.09e4 + 5.35e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.28e4 + 5.69e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.01e4T + 3.93e9T^{2} \)
89 \( 1 + (3.98e3 - 6.90e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.43e5T + 8.58e9T^{2} \)
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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

−11.64733014282417172743953315351, −11.14144308183363609039034812502, −9.753670801203777950404020468417, −8.627453960217855855492001282956, −7.46438828438612236976851642847, −5.79252415889089787462075212245, −4.96153111397304907339849404279, −3.39267164074515454461129413462, −1.89983717481718599690687647386, −0.799333639744610196393175374369, 1.98847966529223393078019735529, 3.75067973866117392600092182719, 5.18963375313436639360770795658, 6.12074945615192284342378821051, 6.90171815360462121997828633759, 8.092374602544204220144592021300, 10.06423054185716363565030336112, 10.23280863615680575985455603828, 11.45110953760180735105289952953, 12.89993140003720604841202311026

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