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Properties

Label 2-140-140.19-c1-0-12
Degree $2$
Conductor $140$
Sign $0.999 + 0.0430i$
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  + (1.41 − 0.0377i)2-s + (−0.634 + 0.366i)3-s + (1.99 − 0.106i)4-s + (1.51 − 1.64i)5-s + (−0.883 + 0.542i)6-s + (−2.56 + 0.664i)7-s + (2.81 − 0.226i)8-s + (−1.23 + 2.13i)9-s + (2.08 − 2.37i)10-s + (2.33 − 1.34i)11-s + (−1.22 + 0.799i)12-s − 3.95·13-s + (−3.59 + 1.03i)14-s + (−0.363 + 1.59i)15-s + (3.97 − 0.426i)16-s + (−0.709 − 1.22i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0267i)2-s + (−0.366 + 0.211i)3-s + (0.998 − 0.0534i)4-s + (0.679 − 0.733i)5-s + (−0.360 + 0.221i)6-s + (−0.967 + 0.250i)7-s + (0.996 − 0.0800i)8-s + (−0.410 + 0.710i)9-s + (0.659 − 0.751i)10-s + (0.702 − 0.405i)11-s + (−0.354 + 0.230i)12-s − 1.09·13-s + (−0.960 + 0.276i)14-s + (−0.0937 + 0.412i)15-s + (0.994 − 0.106i)16-s + (−0.172 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0430i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.999 + 0.0430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72003 - 0.0370256i\)
\(L(\frac12)\) \(\approx\) \(1.72003 - 0.0370256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0377i)T \)
5 \( 1 + (-1.51 + 1.64i)T \)
7 \( 1 + (2.56 - 0.664i)T \)
good3 \( 1 + (0.634 - 0.366i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.95T + 13T^{2} \)
17 \( 1 + (0.709 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.45 - 4.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 + (3.81 + 6.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.87 - 2.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.325iT - 41T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + (-5.68 - 3.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.39 - 0.807i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.81 - 6.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.3 - 7.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.51 + 2.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 + (-0.709 - 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 + 6.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.26iT - 83T^{2} \)
89 \( 1 + (-4.10 - 2.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.35T + 97T^{2} \)
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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

−13.16308053430735941153455405633, −12.29876281222083037862680944251, −11.41571293802421001297967592628, −10.15503235805543245485251983834, −9.237720465103386743518702488210, −7.58887008233114914482240632905, −6.07936529893881130141338984968, −5.49193118206019787212516390101, −4.12699152617682515560503726357, −2.36063203162082806064459962495, 2.49551075207156477461526679165, 3.88540301008099137523829466337, 5.58508706212703945597423617547, 6.60146150200118903990510738026, 7.10264539127212407319223722067, 9.273432445908771940935673093695, 10.30001968970007435801307642678, 11.32627398023006863320927966094, 12.44068675116378996254075135923, 12.98207131246802387525385853357

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