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Properties

Label 2-140-35.12-c1-0-2
Degree $2$
Conductor $140$
Sign $0.788 + 0.614i$
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  + (0.409 − 1.52i)3-s + (1.62 + 1.54i)5-s + (−0.512 − 2.59i)7-s + (0.434 + 0.250i)9-s + (−0.342 − 0.593i)11-s + (−1.04 − 1.04i)13-s + (3.01 − 1.84i)15-s + (−1.31 − 0.353i)17-s + (−1.55 + 2.70i)19-s + (−4.17 − 0.280i)21-s + (1.18 + 4.44i)23-s + (0.251 + 4.99i)25-s + (3.91 − 3.91i)27-s + 7.90i·29-s + (−7.63 + 4.40i)31-s + ⋯
L(s)  = 1  + (0.236 − 0.881i)3-s + (0.724 + 0.689i)5-s + (−0.193 − 0.981i)7-s + (0.144 + 0.0835i)9-s + (−0.103 − 0.178i)11-s + (−0.290 − 0.290i)13-s + (0.778 − 0.476i)15-s + (−0.320 − 0.0857i)17-s + (−0.357 + 0.619i)19-s + (−0.910 − 0.0611i)21-s + (0.248 + 0.925i)23-s + (0.0503 + 0.998i)25-s + (0.753 − 0.753i)27-s + 1.46i·29-s + (−1.37 + 0.791i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.788 + 0.614i$
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: \((2,\ 140,\ (\ :1/2),\ 0.788 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19526 - 0.410880i\)
\(L(\frac12)\) \(\approx\) \(1.19526 - 0.410880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.54i)T \)
7 \( 1 + (0.512 + 2.59i)T \)
good3 \( 1 + (-0.409 + 1.52i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.342 + 0.593i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.04 + 1.04i)T + 13iT^{2} \)
17 \( 1 + (1.31 + 0.353i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.55 - 2.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.18 - 4.44i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.90iT - 29T^{2} \)
31 \( 1 + (7.63 - 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.68 - 2.32i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-3.73 + 3.73i)T - 43iT^{2} \)
47 \( 1 + (1.80 + 6.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.94 - 2.39i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.10 - 3.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.57 - 2.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.07 - 4.01i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (-3.32 + 12.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (12.1 + 7.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.99 + 3.99i)T + 83iT^{2} \)
89 \( 1 + (-5.79 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.60 + 2.60i)T - 97iT^{2} \)
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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

−13.23952192161141210165525898385, −12.30810939133370217237772338096, −10.75629054780988802174341544967, −10.22934644891176650774528523747, −8.813950867587028215239787722575, −7.28919772622864229688325075393, −6.96090255387356319340488262648, −5.44083138975188008005252238402, −3.48936879897547103803596212441, −1.79432180712546749212748829072, 2.38318590736236426961807463603, 4.26578143350511669779678721692, 5.32105256410988705582551270379, 6.58007283843273246421263275356, 8.403111733916133537915790808127, 9.317064861603989216676330781851, 9.835812530476524182792653399105, 11.14150591710826410018990361327, 12.45704885269468817489972732940, 13.09934921172554104023276814015

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