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Properties

Label 2-140-28.3-c1-0-12
Degree $2$
Conductor $140$
Sign $0.998 - 0.0510i$
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  + (0.836 + 1.14i)2-s + (1.51 − 2.62i)3-s + (−0.601 + 1.90i)4-s + (0.866 − 0.5i)5-s + (4.25 − 0.465i)6-s + (−2.57 + 0.602i)7-s + (−2.67 + 0.908i)8-s + (−3.08 − 5.33i)9-s + (1.29 + 0.569i)10-s + (1.03 + 0.598i)11-s + (4.08 + 4.46i)12-s + 4.83i·13-s + (−2.84 − 2.43i)14-s − 3.02i·15-s + (−3.27 − 2.29i)16-s + (2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.591 + 0.806i)2-s + (0.873 − 1.51i)3-s + (−0.300 + 0.953i)4-s + (0.387 − 0.223i)5-s + (1.73 − 0.190i)6-s + (−0.973 + 0.227i)7-s + (−0.946 + 0.321i)8-s + (−1.02 − 1.77i)9-s + (0.409 + 0.180i)10-s + (0.312 + 0.180i)11-s + (1.18 + 1.28i)12-s + 1.34i·13-s + (−0.759 − 0.650i)14-s − 0.781i·15-s + (−0.819 − 0.573i)16-s + (0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.998 - 0.0510i$
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: \((2,\ 140,\ (\ :1/2),\ 0.998 - 0.0510i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68904 + 0.0431474i\)
\(L(\frac12)\) \(\approx\) \(1.68904 + 0.0431474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.836 - 1.14i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.57 - 0.602i)T \)
good3 \( 1 + (-1.51 + 2.62i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.03 - 0.598i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.83iT - 13T^{2} \)
17 \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.711 + 1.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.02 - 2.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.774T + 29T^{2} \)
31 \( 1 + (-3.31 + 5.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.55 + 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.46iT - 41T^{2} \)
43 \( 1 + 1.38iT - 43T^{2} \)
47 \( 1 + (-0.535 - 0.927i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.68 - 2.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.94 - 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.31 + 4.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.14 + 5.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (-0.0927 - 0.0535i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.32 + 5.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (3.41 - 1.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.71iT - 97T^{2} \)
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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

−13.46139005442659148763443990847, −12.42720423433202367646731669717, −11.93894575254801000486065122150, −9.491288617156262209965605468748, −8.754876362646814371150583711645, −7.59220977332708715184127812881, −6.68955398333934368313667063265, −5.93092431674634112284115886856, −3.81693715062581311786679163688, −2.27682159207375784420897951913, 2.87387437052319920157967322820, 3.58780630399374764527063579619, 4.91852848930117050261153234535, 6.18172008516647596428480056046, 8.328498344095654982623586675796, 9.528011266322131514386731159238, 10.13169854678544455357345304289, 10.68076238876206467623859346615, 12.19917434054029476535792882017, 13.35280209327292798397472976634

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