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Properties

Label 2-140-28.3-c1-0-10
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: $\chi_{140} (31, \cdot )$
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

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

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