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Properties

Label 2-140-140.67-c1-0-2
Degree $2$
Conductor $140$
Sign $-0.369 - 0.929i$
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  + (−1.05 + 0.942i)2-s + (−1.20 + 0.322i)3-s + (0.223 − 1.98i)4-s + (2.09 + 0.771i)5-s + (0.965 − 1.47i)6-s + (−0.451 + 2.60i)7-s + (1.63 + 2.30i)8-s + (−1.25 + 0.722i)9-s + (−2.93 + 1.16i)10-s + (0.824 + 0.476i)11-s + (0.372 + 2.46i)12-s + (−3.15 + 3.15i)13-s + (−1.98 − 3.17i)14-s + (−2.77 − 0.251i)15-s + (−3.90 − 0.886i)16-s + (2.56 − 0.688i)17-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + (−0.695 + 0.186i)3-s + (0.111 − 0.993i)4-s + (0.938 + 0.344i)5-s + (0.394 − 0.602i)6-s + (−0.170 + 0.985i)7-s + (0.579 + 0.815i)8-s + (−0.417 + 0.240i)9-s + (−0.929 + 0.368i)10-s + (0.248 + 0.143i)11-s + (0.107 + 0.711i)12-s + (−0.875 + 0.875i)13-s + (−0.529 − 0.848i)14-s + (−0.716 − 0.0649i)15-s + (−0.975 − 0.221i)16-s + (0.623 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.369 - 0.929i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361625 + 0.532883i\)
\(L(\frac12)\) \(\approx\) \(0.361625 + 0.532883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 - 0.942i)T \)
5 \( 1 + (-2.09 - 0.771i)T \)
7 \( 1 + (0.451 - 2.60i)T \)
good3 \( 1 + (1.20 - 0.322i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.824 - 0.476i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.15 - 3.15i)T - 13iT^{2} \)
17 \( 1 + (-2.56 + 0.688i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.99 - 3.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.141 - 0.528i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.23iT - 29T^{2} \)
31 \( 1 + (-3.41 - 1.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.269 + 1.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71T + 41T^{2} \)
43 \( 1 + (8.73 + 8.73i)T + 43iT^{2} \)
47 \( 1 + (-3.09 - 0.830i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.760 + 2.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.30 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.01 - 5.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.85 + 14.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.60iT - 71T^{2} \)
73 \( 1 + (-3.80 - 14.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.34 - 9.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.72 - 5.72i)T + 83iT^{2} \)
89 \( 1 + (1.83 - 1.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.08 + 3.08i)T + 97iT^{2} \)
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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

−13.87235539374097653535546827445, −12.16836489214982482174709578689, −11.34372702146334476818379453605, −10.03843633356636225024916421632, −9.534859151162945179498005147903, −8.298167251975728771007680880092, −6.83346369588115869647802047074, −5.87305870206045801227707831865, −5.13416749289404243603094074806, −2.23883141740540163168491182021, 0.942243420797544293005249396264, 3.04949509969607683996026973535, 4.96481288952876415502388095379, 6.41484370501396047783708247294, 7.56312127235886491487618842888, 8.926230254167433408403516269921, 9.955812960662915360833750008800, 10.64639875277784816686158851899, 11.75268805272258339540386847944, 12.67830025053639598535062038540

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