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Properties

Label 2-133-7.4-c1-0-3
Degree $2$
Conductor $133$
Sign $-0.749 - 0.661i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
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  + (1.20 + 2.09i)2-s + (−0.707 + 1.22i)3-s + (−1.91 + 3.31i)4-s + (−0.5 − 0.866i)5-s − 3.41·6-s + (1.62 − 2.09i)7-s − 4.41·8-s + (0.500 + 0.866i)9-s + (1.20 − 2.09i)10-s + (0.207 − 0.358i)11-s + (−2.70 − 4.68i)12-s − 2.24·13-s + (6.32 + 0.866i)14-s + 1.41·15-s + (−1.49 − 2.59i)16-s + (2 − 3.46i)17-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.408 + 0.707i)3-s + (−0.957 + 1.65i)4-s + (−0.223 − 0.387i)5-s − 1.39·6-s + (0.612 − 0.790i)7-s − 1.56·8-s + (0.166 + 0.288i)9-s + (0.381 − 0.661i)10-s + (0.0624 − 0.108i)11-s + (−0.781 − 1.35i)12-s − 0.621·13-s + (1.69 + 0.231i)14-s + 0.365·15-s + (−0.374 − 0.649i)16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498375 + 1.31809i\)
\(L(\frac12)\) \(\approx\) \(0.498375 + 1.31809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.62 + 2.09i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.207 + 0.358i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.79 - 4.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.58T + 29T^{2} \)
31 \( 1 + (-3.12 + 5.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.53 + 7.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.17T + 41T^{2} \)
43 \( 1 - 8.07T + 43T^{2} \)
47 \( 1 + (-2.20 - 3.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.41 - 5.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.91 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.585 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 + (7.57 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.707 + 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.24T + 83T^{2} \)
89 \( 1 + (-5.70 - 9.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.17T + 97T^{2} \)
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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

−13.90440495248938702468937245261, −13.03417695314521917518795868456, −11.79578091488909985602502555359, −10.62734739923770138590321754395, −9.304598682106100222203772593334, −7.80194237915347831012503672633, −7.26981639891202818611628110860, −5.61032274618736934319138033502, −4.82829740307763781934580890757, −3.94320343687399321611910970285, 1.62349896887612359264536673210, 3.11143406574674225220848696824, 4.67084015104798149042497717611, 5.86351719539445935547290385186, 7.29923213739200502502863414631, 8.945936579556195760899228822316, 10.26904842344517815902100094184, 11.18297410856701043647766214653, 12.10876596677178615245203444286, 12.45152574158557159197484511432

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