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Properties

Label 2-140-140.59-c1-0-12
Degree $2$
Conductor $140$
Sign $0.557 - 0.830i$
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  + (1.13 + 0.843i)2-s + (1.62 + 0.940i)3-s + (0.576 + 1.91i)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 + (−0.672 − 3.08i)10-s + (1.20 + 0.692i)11-s + (−0.861 + 3.66i)12-s − 5.79·13-s + (1.48 − 3.43i)14-s + (−1.36 − 3.97i)15-s + (−3.33 + 2.20i)16-s + (2.33 − 4.04i)17-s + ⋯
L(s)  = 1  + (0.802 + 0.596i)2-s + (0.940 + 0.542i)3-s + (0.288 + 0.957i)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 + (−0.212 − 0.977i)10-s + (0.361 + 0.208i)11-s + (−0.248 + 1.05i)12-s − 1.60·13-s + (0.397 − 0.917i)14-s + (−0.351 − 1.02i)15-s + (−0.833 + 0.552i)16-s + (0.565 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.557 - 0.830i$
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: \((2,\ 140,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60239 + 0.854076i\)
\(L(\frac12)\) \(\approx\) \(1.60239 + 0.854076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.843i)T \)
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

−13.63373505606641911870373989129, −12.36747376693842056951251025866, −11.75222203974096209729631470645, −9.991828738981388941924588119981, −9.032583838410371270101785799646, −7.74987449202283619113578143005, −7.19253700247045083855709420473, −5.20083492084338180342173321758, −4.12332186093855076112035059809, −3.18643146249361302528375746332, 2.41666487274403083838941794956, 3.22357072026904277251843295618, 4.91782664823083381107598201351, 6.49704965917582067094879477712, 7.58426913686068039163979996666, 8.847153438039891635056541399153, 10.01972200262566472551930571822, 11.29777654030366932146557546974, 12.16535226538556345272127029361, 12.89194885595829873671670047582

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