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Properties

Label 2-140-140.123-c1-0-6
Degree $2$
Conductor $140$
Sign $-0.591 - 0.806i$
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  + (0.0671 + 1.41i)2-s + (0.638 + 2.38i)3-s + (−1.99 + 0.189i)4-s + (2.14 + 0.631i)5-s + (−3.32 + 1.06i)6-s + (1.60 − 2.10i)7-s + (−0.401 − 2.79i)8-s + (−2.68 + 1.54i)9-s + (−0.748 + 3.07i)10-s + (−4.09 − 2.36i)11-s + (−1.72 − 4.62i)12-s + (0.0592 + 0.0592i)13-s + (3.07 + 2.13i)14-s + (−0.135 + 5.51i)15-s + (3.92 − 0.755i)16-s + (−1.27 − 4.77i)17-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (0.368 + 1.37i)3-s + (−0.995 + 0.0948i)4-s + (0.959 + 0.282i)5-s + (−1.35 + 0.433i)6-s + (0.607 − 0.794i)7-s + (−0.142 − 0.989i)8-s + (−0.893 + 0.515i)9-s + (−0.236 + 0.971i)10-s + (−1.23 − 0.713i)11-s + (−0.497 − 1.33i)12-s + (0.0164 + 0.0164i)13-s + (0.822 + 0.569i)14-s + (−0.0349 + 1.42i)15-s + (0.981 − 0.188i)16-s + (−0.310 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.591 - 0.806i$
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: \((2,\ 140,\ (\ :1/2),\ -0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578023 + 1.14094i\)
\(L(\frac12)\) \(\approx\) \(0.578023 + 1.14094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0671 - 1.41i)T \)
5 \( 1 + (-2.14 - 0.631i)T \)
7 \( 1 + (-1.60 + 2.10i)T \)
good3 \( 1 + (-0.638 - 2.38i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0592 - 0.0592i)T + 13iT^{2} \)
17 \( 1 + (1.27 + 4.77i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.31 - 2.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.09 + 0.292i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.27iT - 29T^{2} \)
31 \( 1 + (4.01 + 2.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.22 - 0.596i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 + (-1.57 + 1.57i)T - 43iT^{2} \)
47 \( 1 + (-0.716 + 2.67i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.12 - 2.44i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.67 + 2.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.978 + 1.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.491 - 0.131i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (-9.26 + 2.48i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.05 + 5.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.62 + 5.62i)T - 83iT^{2} \)
89 \( 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.81 + 5.81i)T - 97iT^{2} \)
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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

−13.97905431573620819378290174603, −13.05362791656311772467199055410, −10.98123971799082588989801910992, −10.21181070087656315641575434682, −9.396976467079560256314866032555, −8.341870148904524462183690448285, −7.12806074405656449407363265381, −5.53129568886183408119182667790, −4.78188790280331617424926420016, −3.31108299589409512519999036216, 1.75766320941467999695530518432, 2.51692578906755829549319084884, 4.91601189338621407517564857015, 6.06747468586898840789259524338, 7.78724860271503223432814786155, 8.597332058790068161903387825789, 9.732716971779046504461252867627, 10.88335213155640200771265041529, 12.14824685342472159196994384538, 12.83715476354109169535563355401

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