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Properties

Label 2-140-140.59-c1-0-19
Degree $2$
Conductor $140$
Sign $-0.794 - 0.607i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (−0.163 − 1.40i)2-s + (−1.62 − 0.940i)3-s + (−1.94 + 0.458i)4-s + (−1.68 − 1.47i)5-s + (−1.05 + 2.44i)6-s + (0.603 + 2.57i)7-s + (0.960 + 2.66i)8-s + (0.267 + 0.463i)9-s + (−1.79 + 2.60i)10-s + (−1.20 − 0.692i)11-s + (3.60 + 1.08i)12-s − 5.79·13-s + (3.52 − 1.26i)14-s + (1.36 + 3.97i)15-s + (3.58 − 1.78i)16-s + (2.33 − 4.04i)17-s + ⋯
L(s)  = 1  + (−0.115 − 0.993i)2-s + (−0.940 − 0.542i)3-s + (−0.973 + 0.229i)4-s + (−0.753 − 0.657i)5-s + (−0.430 + 0.996i)6-s + (0.228 + 0.973i)7-s + (0.339 + 0.940i)8-s + (0.0891 + 0.154i)9-s + (−0.566 + 0.824i)10-s + (−0.361 − 0.208i)11-s + (1.03 + 0.313i)12-s − 1.60·13-s + (0.940 − 0.338i)14-s + (0.351 + 1.02i)15-s + (0.895 − 0.445i)16-s + (0.565 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.794 - 0.607i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0961570 + 0.284031i\)
\(L(\frac12)\) \(\approx\) \(0.0961570 + 0.284031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.163 + 1.40i)T \)
5 \( 1 + (1.68 + 1.47i)T \)
7 \( 1 + (-0.603 - 2.57i)T \)
good3 \( 1 + (1.62 + 0.940i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.20 + 0.692i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.79T + 13T^{2} \)
17 \( 1 + (-2.33 + 4.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.74 + 4.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.04T + 29T^{2} \)
31 \( 1 + (-0.832 + 1.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.247 - 0.142i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.21iT - 41T^{2} \)
43 \( 1 - 3.30T + 43T^{2} \)
47 \( 1 + (-0.119 + 0.0690i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.82 - 3.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.832 - 1.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.49 - 1.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.82 + 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.29iT - 71T^{2} \)
73 \( 1 + (2.33 - 4.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.88 - 5.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.0814iT - 83T^{2} \)
89 \( 1 + (3.22 - 1.85i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.88T + 97T^{2} \)
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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

−12.15660441897745849636036748986, −11.90271562306414471798328984945, −10.87589136140804995048744382191, −9.476795974288872478339565149266, −8.517370156257241018997905248101, −7.29847986605777983098649073438, −5.47594071716603713994669607598, −4.66556537509150262193044323046, −2.59749333982331659422497430426, −0.34611432059018200139288731262, 3.95312121420654334453369607135, 4.92926293196669225164354649440, 6.19978364568557463645230798535, 7.43266633089005818259954943664, 8.061226527930424063280730753240, 10.15063180184454835613457387543, 10.28268665595349689290749637240, 11.66436506624502845738388595232, 12.74298367178618259419816217533, 14.21802440918870075896184970491

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