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Properties

Label 2-140-28.19-c1-0-1
Degree $2$
Conductor $140$
Sign $-0.315 - 0.948i$
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  + (−0.894 + 1.09i)2-s + (0.895 + 1.55i)3-s + (−0.400 − 1.95i)4-s + (0.866 + 0.5i)5-s + (−2.49 − 0.405i)6-s + (0.644 + 2.56i)7-s + (2.50 + 1.31i)8-s + (−0.103 + 0.179i)9-s + (−1.32 + 0.501i)10-s + (−3.66 + 2.11i)11-s + (2.68 − 2.37i)12-s − 2.98i·13-s + (−3.38 − 1.58i)14-s + 1.79i·15-s + (−3.67 + 1.56i)16-s + (−1.92 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (0.516 + 0.895i)3-s + (−0.200 − 0.979i)4-s + (0.387 + 0.223i)5-s + (−1.02 − 0.165i)6-s + (0.243 + 0.969i)7-s + (0.885 + 0.464i)8-s + (−0.0344 + 0.0596i)9-s + (−0.418 + 0.158i)10-s + (−1.10 + 0.638i)11-s + (0.773 − 0.685i)12-s − 0.827i·13-s + (−0.905 − 0.424i)14-s + 0.462i·15-s + (−0.919 + 0.392i)16-s + (−0.465 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.315 - 0.948i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.568998 + 0.788881i\)
\(L(\frac12)\) \(\approx\) \(0.568998 + 0.788881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.894 - 1.09i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.644 - 2.56i)T \)
good3 \( 1 + (-0.895 - 1.55i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.66 - 2.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.98iT - 13T^{2} \)
17 \( 1 + (1.92 - 1.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.28 + 3.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.78 - 1.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + (1.20 + 2.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.16 + 3.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.88iT - 41T^{2} \)
43 \( 1 + 12.3iT - 43T^{2} \)
47 \( 1 + (3.38 - 5.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.99 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0195 - 0.0113i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.38 - 2.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.07iT - 71T^{2} \)
73 \( 1 + (2.88 - 1.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.14 - 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (14.4 + 8.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−13.80208991506116356619844562341, −12.59968150904733212344538060406, −10.92571672588507761637133755640, −10.10857875593361727900820391803, −9.247882985265117932467656175812, −8.431975560894438291878959450769, −7.20333046220292119903713018370, −5.71323480772660484826340791170, −4.76004323732188254405498747857, −2.61285392541879979360966646381, 1.39485870785503188030160246824, 2.88481934920575227921112294070, 4.65201271949668256164780137038, 6.77456227758837316164525397097, 7.82201319986103485435414033515, 8.508320670773123945711291614668, 9.871066643321713611189790779731, 10.73191163987918856941452736544, 11.81400084467743872272970934649, 13.05310168605596949699451481380

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