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Properties

Label 2-147-147.62-c1-0-1
Degree $2$
Conductor $147$
Sign $0.0928 - 0.995i$
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  + (0.500 − 0.114i)2-s + (−1.71 − 0.247i)3-s + (−1.56 + 0.753i)4-s + (1.88 + 2.36i)5-s + (−0.886 + 0.0716i)6-s + (−1.31 + 2.29i)7-s + (−1.49 + 1.19i)8-s + (2.87 + 0.850i)9-s + (1.21 + 0.966i)10-s + (1.61 − 0.367i)11-s + (2.86 − 0.903i)12-s + (−2.41 + 0.551i)13-s + (−0.395 + 1.29i)14-s + (−2.64 − 4.51i)15-s + (1.55 − 1.94i)16-s + (1.01 + 0.488i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.0807i)2-s + (−0.989 − 0.143i)3-s + (−0.782 + 0.376i)4-s + (0.842 + 1.05i)5-s + (−0.361 + 0.0292i)6-s + (−0.496 + 0.868i)7-s + (−0.530 + 0.422i)8-s + (0.959 + 0.283i)9-s + (0.383 + 0.305i)10-s + (0.485 − 0.110i)11-s + (0.828 − 0.260i)12-s + (−0.670 + 0.153i)13-s + (−0.105 + 0.347i)14-s + (−0.682 − 1.16i)15-s + (0.387 − 0.486i)16-s + (0.246 + 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.0928 - 0.995i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.0928 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615876 + 0.561110i\)
\(L(\frac12)\) \(\approx\) \(0.615876 + 0.561110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.247i)T \)
7 \( 1 + (1.31 - 2.29i)T \)
good2 \( 1 + (-0.500 + 0.114i)T + (1.80 - 0.867i)T^{2} \)
5 \( 1 + (-1.88 - 2.36i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-1.61 + 0.367i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (2.41 - 0.551i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-1.01 - 0.488i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 5.26iT - 19T^{2} \)
23 \( 1 + (2.57 + 5.35i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-2.83 + 5.88i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + 1.14iT - 31T^{2} \)
37 \( 1 + (-10.9 - 5.26i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-7.40 - 9.28i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-0.729 + 0.914i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.915 - 4.01i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.04 - 2.16i)T + (-33.0 + 41.4i)T^{2} \)
59 \( 1 + (-6.97 + 8.74i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (1.46 - 3.03i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 6.56T + 67T^{2} \)
71 \( 1 + (0.105 + 0.219i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.70 + 1.52i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 + (1.61 - 7.05i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-1.66 + 7.29i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 9.63iT - 97T^{2} \)
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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

−13.12265689699304256433857491935, −12.32092599211036722332687275030, −11.54615391932206418286053799482, −10.11333090985792273156488306756, −9.575204203490919790376440687259, −7.963482811399925275932787604233, −6.34365661423551892736554759395, −5.87764846138382453651070873160, −4.37824042519894261641766351724, −2.64012279293848123617524607648, 0.906489026859686325676303116616, 4.11423285868203239959145410142, 5.06891642307698586286790927854, 5.91836011338173899797389264218, 7.19109672037999613052211144098, 9.169086959223365374276921862950, 9.655840729737360359530330668728, 10.65136106288182909282719285224, 12.10695618546327835047330812278, 12.96703858182401952162576574596

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