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Properties

Label 2-140-35.12-c1-0-3
Degree $2$
Conductor $140$
Sign $0.0480 + 0.998i$
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  + (0.649 − 2.42i)3-s + (−0.746 − 2.10i)5-s + (−0.513 + 2.59i)7-s + (−2.86 − 1.65i)9-s + (−1.86 − 3.22i)11-s + (2.90 + 2.90i)13-s + (−5.59 + 0.440i)15-s + (6.67 + 1.78i)17-s + (1.65 − 2.86i)19-s + (5.96 + 2.93i)21-s + (0.493 + 1.84i)23-s + (−3.88 + 3.14i)25-s + (−0.539 + 0.539i)27-s + 0.563i·29-s + (−3.20 + 1.85i)31-s + ⋯
L(s)  = 1  + (0.375 − 1.40i)3-s + (−0.333 − 0.942i)5-s + (−0.194 + 0.980i)7-s + (−0.953 − 0.550i)9-s + (−0.561 − 0.973i)11-s + (0.805 + 0.805i)13-s + (−1.44 + 0.113i)15-s + (1.61 + 0.433i)17-s + (0.380 − 0.658i)19-s + (1.30 + 0.639i)21-s + (0.102 + 0.383i)23-s + (−0.777 + 0.629i)25-s + (−0.103 + 0.103i)27-s + 0.104i·29-s + (−0.575 + 0.332i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0480 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0480 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.0480 + 0.998i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.0480 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833291 - 0.794159i\)
\(L(\frac12)\) \(\approx\) \(0.833291 - 0.794159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.746 + 2.10i)T \)
7 \( 1 + (0.513 - 2.59i)T \)
good3 \( 1 + (-0.649 + 2.42i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.86 + 3.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 - 2.90i)T + 13iT^{2} \)
17 \( 1 + (-6.67 - 1.78i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.65 + 2.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.493 - 1.84i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.563iT - 29T^{2} \)
31 \( 1 + (3.20 - 1.85i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.00 + 0.268i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.88iT - 41T^{2} \)
43 \( 1 + (7.53 - 7.53i)T - 43iT^{2} \)
47 \( 1 + (0.145 + 0.543i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.78 + 0.479i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.38 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.00 - 4.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.49 - 5.56i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.683T + 71T^{2} \)
73 \( 1 + (-3.11 + 11.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.76 - 1.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.44 + 1.44i)T + 83iT^{2} \)
89 \( 1 + (-1.26 + 2.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.99 - 3.99i)T - 97iT^{2} \)
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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

−12.96483846928224532528447135835, −12.10691128188596701427107000070, −11.34107820036569392627712436406, −9.434675923237602351568166136522, −8.434513975133149092489111093395, −7.86059845502820897240422314440, −6.38596359891587516662575072547, −5.33094624098176158010183878443, −3.20506720343788994783101499864, −1.40374522649104772401200128194, 3.19246395023333743744005671653, 3.98358124401562832563600368529, 5.44101516212366785643537381418, 7.16738764065858122236346872762, 8.088243076754115561699083016867, 9.740328935458027706139946809181, 10.26157665552234484518461255424, 10.90132838115912433664411484587, 12.32218189215265140974028148366, 13.72153010938711788679034917549

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