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Properties

Label 2-147-147.101-c1-0-10
Degree $2$
Conductor $147$
Sign $0.914 + 0.403i$
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  + (0.742 + 0.291i)2-s + (0.844 − 1.51i)3-s + (−0.999 − 0.927i)4-s + (3.49 + 2.38i)5-s + (1.06 − 0.876i)6-s + (−2.63 − 0.242i)7-s + (−1.16 − 2.41i)8-s + (−1.57 − 2.55i)9-s + (1.90 + 2.78i)10-s + (0.481 + 3.19i)11-s + (−2.24 + 0.727i)12-s + (1.47 − 1.17i)13-s + (−1.88 − 0.947i)14-s + (6.55 − 3.27i)15-s + (0.0439 + 0.586i)16-s + (−0.330 − 0.101i)17-s + ⋯
L(s)  = 1  + (0.524 + 0.206i)2-s + (0.487 − 0.872i)3-s + (−0.499 − 0.463i)4-s + (1.56 + 1.06i)5-s + (0.435 − 0.357i)6-s + (−0.995 − 0.0917i)7-s + (−0.411 − 0.854i)8-s + (−0.524 − 0.851i)9-s + (0.600 + 0.881i)10-s + (0.145 + 0.963i)11-s + (−0.648 + 0.210i)12-s + (0.410 − 0.327i)13-s + (−0.503 − 0.253i)14-s + (1.69 − 0.844i)15-s + (0.0109 + 0.146i)16-s + (−0.0801 − 0.0247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56161 - 0.329149i\)
\(L(\frac12)\) \(\approx\) \(1.56161 - 0.329149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.844 + 1.51i)T \)
7 \( 1 + (2.63 + 0.242i)T \)
good2 \( 1 + (-0.742 - 0.291i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (-3.49 - 2.38i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.481 - 3.19i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.47 + 1.17i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.330 + 0.101i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (3.79 - 2.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.617 - 2.00i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (3.02 - 0.690i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.61 + 3.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.19 + 3.88i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-5.35 + 2.58i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.13 + 0.545i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.205 - 0.523i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (0.0176 - 0.0190i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-2.74 + 1.87i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (4.20 + 4.53i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (-6.45 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.01 - 1.60i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.47 - 0.972i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (1.49 + 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.31 + 6.66i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.88 + 0.284i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−13.03374277477196109200151249889, −12.75405713080069043083692174264, −10.76757535847554785946841871617, −9.707561947587011168033473133296, −9.217492453926486480611222021817, −7.24644725104330220106645028879, −6.34620297298104769727044824117, −5.74260319161519100005625326063, −3.56920722644674155297767972073, −2.06533037920862605213729691606, 2.62558687860289765025031630144, 4.01487436891207547417813785423, 5.21861274803070097193901292554, 6.14341680075367651153248551982, 8.545166922485943664577632889437, 9.012211126927867068647327062224, 9.773224546239354613394821583299, 11.02828593510534961799654757757, 12.59698082161247324244190540751, 13.28494360636994532382527317988

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