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Properties

Label 2-140-140.123-c1-0-12
Degree $2$
Conductor $140$
Sign $-0.290 + 0.956i$
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  + (−1.36 − 0.366i)2-s + (−0.684 − 2.55i)3-s + (1.73 + i)4-s + (2.23 − 0.133i)5-s + 3.74i·6-s + (2.55 − 0.684i)7-s + (−1.99 − 2i)8-s + (−3.46 + 2i)9-s + (−3.09 − 0.633i)10-s + (−3.24 − 1.87i)11-s + (1.36 − 5.11i)12-s + (−2 − 2i)13-s − 3.74·14-s + (−1.87 − 5.61i)15-s + (1.99 + 3.46i)16-s + (0.732 + 2.73i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.395 − 1.47i)3-s + (0.866 + 0.5i)4-s + (0.998 − 0.0599i)5-s + 1.52i·6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−1.15 + 0.666i)9-s + (−0.979 − 0.200i)10-s + (−0.977 − 0.564i)11-s + (0.395 − 1.47i)12-s + (−0.554 − 0.554i)13-s − 0.999·14-s + (−0.483 − 1.44i)15-s + (0.499 + 0.866i)16-s + (0.177 + 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455603 - 0.614743i\)
\(L(\frac12)\) \(\approx\) \(0.455603 - 0.614743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
5 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 + (-2.55 + 0.684i)T \)
good3 \( 1 + (0.684 + 2.55i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (3.24 + 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (-0.732 - 2.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.87 - 3.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.55 + 0.684i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-6.48 - 3.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + (5.61 - 5.61i)T - 43iT^{2} \)
47 \( 1 + (2.73 - 10.2i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.87 + 3.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 3.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.74iT - 71T^{2} \)
73 \( 1 + (2.73 - 0.732i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.87 + 3.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.87 + 1.87i)T - 83iT^{2} \)
89 \( 1 + (-2.59 + 1.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9 - 9i)T - 97iT^{2} \)
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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

−12.71902165271324692558173851859, −11.85892135071946051152944431785, −10.77279332468127952106955467818, −9.937472672246282540460329447193, −8.226434844781075043010928086744, −7.82951480732787418970521506538, −6.51300917382740182592681028660, −5.51194099626312332973526697527, −2.49626399535313648478583362581, −1.23768073943128594321233979278, 2.39170931242141706673947769068, 4.88289261762142360257596404115, 5.50678510114889085673083288254, 7.12000321461546355646656480952, 8.555584754327484171778608734256, 9.628751132180991133568023957868, 10.09617740650104818679094905753, 11.04236771730457519287201789246, 11.92593353113647343764907197689, 13.80059337005003233896651176872

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