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Properties

Label 2-147-7.4-c5-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5 − 8.66i)2-s + (4.5 − 7.79i)3-s + (−34.0 + 58.8i)4-s + (−53 − 91.7i)5-s − 90.0·6-s + 360.·8-s + (−40.5 − 70.1i)9-s + (−530. + 917. i)10-s + (−46 + 79.6i)11-s + (306. + 530. i)12-s − 670·13-s − 954·15-s + (−712. − 1.23e3i)16-s + (−111 + 192. i)17-s + (−405. + 701. i)18-s + (−454 − 786. i)19-s + ⋯
L(s)  = 1  + (−0.883 − 1.53i)2-s + (0.288 − 0.499i)3-s + (−1.06 + 1.84i)4-s + (−0.948 − 1.64i)5-s − 1.02·6-s + 1.98·8-s + (−0.166 − 0.288i)9-s + (−1.67 + 2.90i)10-s + (−0.114 + 0.198i)11-s + (0.613 + 1.06i)12-s − 1.09·13-s − 1.09·15-s + (−0.695 − 1.20i)16-s + (−0.0931 + 0.161i)17-s + (−0.294 + 0.510i)18-s + (−0.288 − 0.499i)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: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + (5 + 8.66i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (53 + 91.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (46 - 79.6i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 670T + 3.71e5T^{2} \)
17 \( 1 + (111 - 192. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (454 + 786. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-588 - 1.01e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.11e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.84e3 + 3.20e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.09e3 + 3.62e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 6.66e3T + 1.15e8T^{2} \)
43 \( 1 + 3.70e3T + 1.47e8T^{2} \)
47 \( 1 + (3.52e3 + 6.11e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.87e4 + 3.25e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.63e4 + 2.83e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (5.40e3 + 9.35e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (3.24e4 - 5.62e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.13e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.94e4 + 3.37e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.40e4 - 7.62e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.18e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.59e4 - 9.68e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.50e5T + 8.58e9T^{2} \)
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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

−11.37315367636394439068735094566, −9.914201976588193898444279984136, −9.033672828147407966409455242586, −8.307225935953399829873509318870, −7.44406504787558047596244551643, −4.92781776686100606455440765326, −3.79680775008331501028716340979, −2.23616474996125501828357020521, −0.939344855949880240400398602966, 0, 2.90818081501560957071977937737, 4.52339498510820275551492125846, 6.09177676295042759808624711549, 7.13330827511479836533571942886, 7.73492990428200221411399017937, 8.786298985494977489643922693219, 10.07165204061110733699915652255, 10.62170409408540556111674976423, 11.94109366001104995738220282935

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