Properties

Label 2-147-147.122-c1-0-2
Degree $2$
Conductor $147$
Sign $-0.968 - 0.248i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.749 + 2.42i)2-s + (−0.787 + 1.54i)3-s + (−3.68 + 2.51i)4-s + (2.16 + 0.326i)5-s + (−4.33 − 0.757i)6-s + (2.09 − 1.61i)7-s + (−4.88 − 3.89i)8-s + (−1.75 − 2.43i)9-s + (0.830 + 5.50i)10-s + (2.76 − 2.98i)11-s + (−0.972 − 7.66i)12-s + (−3.59 − 0.821i)13-s + (5.48 + 3.88i)14-s + (−2.21 + 3.08i)15-s + (2.54 − 6.48i)16-s + (−0.243 + 3.24i)17-s + ⋯
L(s)  = 1  + (0.529 + 1.71i)2-s + (−0.454 + 0.890i)3-s + (−1.84 + 1.25i)4-s + (0.968 + 0.146i)5-s + (−1.77 − 0.309i)6-s + (0.792 − 0.609i)7-s + (−1.72 − 1.37i)8-s + (−0.586 − 0.810i)9-s + (0.262 + 1.74i)10-s + (0.834 − 0.899i)11-s + (−0.280 − 2.21i)12-s + (−0.997 − 0.227i)13-s + (1.46 + 1.03i)14-s + (−0.570 + 0.796i)15-s + (0.636 − 1.62i)16-s + (−0.0590 + 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166989 + 1.32350i\)
\(L(\frac12)\) \(\approx\) \(0.166989 + 1.32350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.787 - 1.54i)T \)
7 \( 1 + (-2.09 + 1.61i)T \)
good2 \( 1 + (-0.749 - 2.42i)T + (-1.65 + 1.12i)T^{2} \)
5 \( 1 + (-2.16 - 0.326i)T + (4.77 + 1.47i)T^{2} \)
11 \( 1 + (-2.76 + 2.98i)T + (-0.822 - 10.9i)T^{2} \)
13 \( 1 + (3.59 + 0.821i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.243 - 3.24i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (2.07 - 1.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.04 + 0.378i)T + (22.7 - 3.42i)T^{2} \)
29 \( 1 + (0.510 + 1.05i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 + (5.91 + 3.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.88 - 4.69i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (4.90 - 6.14i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (5.60 + 7.03i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.58 + 0.797i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (-0.0703 - 0.103i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-7.17 + 1.08i)T + (56.3 - 17.3i)T^{2} \)
61 \( 1 + (2.82 - 4.14i)T + (-22.2 - 56.7i)T^{2} \)
67 \( 1 + (-6.74 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.53 - 5.27i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.14 - 3.71i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (-0.648 - 1.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.56 + 6.87i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (5.41 - 5.02i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + 3.58iT - 97T^{2} \)
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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

−13.94438284371558821145967513161, −13.00312005388756956207519024295, −11.50121892117042478632686517842, −10.28336081438419500529995619657, −9.183883021660999114594409808718, −8.164804309797241070153016562903, −6.76403907002461115163858110627, −5.87231110366646233497202496181, −4.97832308544841502763560217762, −3.85829075090126372933750063794, 1.61542140576214163578930512096, 2.45845840757229439172948414089, 4.71346343790784054931095879216, 5.48552512676105422006720027651, 7.08359355881458064734283935666, 8.953329163163564345646767443111, 9.723476976548904626911732021769, 10.98956547699217135243957260610, 11.73847555546678884733119059608, 12.47678943326415088717204347102

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