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Properties

Label 2-140-140.123-c1-0-15
Degree $2$
Conductor $140$
Sign $0.590 + 0.806i$
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  + (0.429 + 1.34i)2-s + (−0.727 − 2.71i)3-s + (−1.63 + 1.15i)4-s + (0.178 − 2.22i)5-s + (3.34 − 2.14i)6-s + (−0.496 − 2.59i)7-s + (−2.25 − 1.70i)8-s + (−4.24 + 2.44i)9-s + (3.08 − 0.715i)10-s + (2.75 + 1.59i)11-s + (4.32 + 3.58i)12-s + (2.41 + 2.41i)13-s + (3.28 − 1.78i)14-s + (−6.18 + 1.13i)15-s + (1.32 − 3.77i)16-s + (−0.600 − 2.24i)17-s + ⋯
L(s)  = 1  + (0.303 + 0.952i)2-s + (−0.419 − 1.56i)3-s + (−0.815 + 0.578i)4-s + (0.0799 − 0.996i)5-s + (1.36 − 0.875i)6-s + (−0.187 − 0.982i)7-s + (−0.798 − 0.601i)8-s + (−1.41 + 0.816i)9-s + (0.974 − 0.226i)10-s + (0.831 + 0.479i)11-s + (1.24 + 1.03i)12-s + (0.670 + 0.670i)13-s + (0.878 − 0.476i)14-s + (−1.59 + 0.293i)15-s + (0.331 − 0.943i)16-s + (−0.145 − 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.590 + 0.806i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.590 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904634 - 0.458788i\)
\(L(\frac12)\) \(\approx\) \(0.904634 - 0.458788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.429 - 1.34i)T \)
5 \( 1 + (-0.178 + 2.22i)T \)
7 \( 1 + (0.496 + 2.59i)T \)
good3 \( 1 + (0.727 + 2.71i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.75 - 1.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \)
17 \( 1 + (0.600 + 2.24i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.39 - 2.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 1.39i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.72iT - 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 0.623i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (3.96 - 3.96i)T - 43iT^{2} \)
47 \( 1 + (-1.63 + 6.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.6 + 3.37i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.951 + 1.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.83 - 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 - 1.51i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.562iT - 71T^{2} \)
73 \( 1 + (3.23 - 0.866i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.13 - 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.38 - 4.38i)T - 83iT^{2} \)
89 \( 1 + (-2.51 + 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−13.29906906314539427009001251890, −12.32407679846750552186694351203, −11.51247146117981531981105871757, −9.513352443891797619239893125021, −8.447856406142634738989914471701, −7.30117495540455819040529581640, −6.71956947617804417599022006687, −5.52279314743794499550279189887, −4.09768188319338784048033319675, −1.14385388712366063806434872252, 2.94114444187696286511911107527, 3.88276415113954334962920504679, 5.34075085402315525397799265238, 6.22137750084434512317596375811, 8.742415664284317655072815043227, 9.460061509012875605676679722121, 10.52951702635236688251795551717, 11.08880581094216448188169731755, 11.87387478249940946344412075468, 13.24144341147804253670470111719

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