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Properties

Label 2-140-20.3-c1-0-16
Degree $2$
Conductor $140$
Sign $-0.343 + 0.939i$
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  + (0.447 − 1.34i)2-s + (0.396 − 0.396i)3-s + (−1.59 − 1.20i)4-s + (−0.137 − 2.23i)5-s + (−0.354 − 0.710i)6-s + (0.707 + 0.707i)7-s + (−2.32 + 1.60i)8-s + 2.68i·9-s + (−3.05 − 0.814i)10-s − 4.30i·11-s + (−1.11 + 0.157i)12-s + (1.27 + 1.27i)13-s + (1.26 − 0.631i)14-s + (−0.940 − 0.831i)15-s + (1.11 + 3.84i)16-s + (−0.355 + 0.355i)17-s + ⋯
L(s)  = 1  + (0.316 − 0.948i)2-s + (0.229 − 0.229i)3-s + (−0.799 − 0.600i)4-s + (−0.0616 − 0.998i)5-s + (−0.144 − 0.289i)6-s + (0.267 + 0.267i)7-s + (−0.823 + 0.567i)8-s + 0.894i·9-s + (−0.966 − 0.257i)10-s − 1.29i·11-s + (−0.320 + 0.0454i)12-s + (0.354 + 0.354i)13-s + (0.338 − 0.168i)14-s + (−0.242 − 0.214i)15-s + (0.278 + 0.960i)16-s + (−0.0862 + 0.0862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722690 - 1.03426i\)
\(L(\frac12)\) \(\approx\) \(0.722690 - 1.03426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.447 + 1.34i)T \)
5 \( 1 + (0.137 + 2.23i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.396 + 0.396i)T - 3iT^{2} \)
11 \( 1 + 4.30iT - 11T^{2} \)
13 \( 1 + (-1.27 - 1.27i)T + 13iT^{2} \)
17 \( 1 + (0.355 - 0.355i)T - 17iT^{2} \)
19 \( 1 - 8.16T + 19T^{2} \)
23 \( 1 + (2.65 - 2.65i)T - 23iT^{2} \)
29 \( 1 - 3.36iT - 29T^{2} \)
31 \( 1 - 0.150iT - 31T^{2} \)
37 \( 1 + (1.53 - 1.53i)T - 37iT^{2} \)
41 \( 1 - 9.17T + 41T^{2} \)
43 \( 1 + (6.48 - 6.48i)T - 43iT^{2} \)
47 \( 1 + (5.35 + 5.35i)T + 47iT^{2} \)
53 \( 1 + (8.37 + 8.37i)T + 53iT^{2} \)
59 \( 1 + 0.357T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (-5.33 - 5.33i)T + 67iT^{2} \)
71 \( 1 + 2.69iT - 71T^{2} \)
73 \( 1 + (4.14 + 4.14i)T + 73iT^{2} \)
79 \( 1 + 9.17T + 79T^{2} \)
83 \( 1 + (4.83 - 4.83i)T - 83iT^{2} \)
89 \( 1 - 1.19iT - 89T^{2} \)
97 \( 1 + (1.04 - 1.04i)T - 97iT^{2} \)
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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

−12.92861227833903113553514699614, −11.73796953054749968515224637365, −11.16641307718963227734301980937, −9.760302305793643074941216946107, −8.755855416846993946121638737387, −7.939466040358891426769919026603, −5.76005364146711025081777806989, −4.86857114370967618987076719120, −3.29908023101509080600131845823, −1.48035516518988031383507427382, 3.20171773153366145934843780020, 4.41732943800133072193617641305, 5.96764935370867789937400730882, 7.07572675865578875030105688181, 7.83616708781710443170677202976, 9.339312102537913475379838215960, 10.10611426667688565435403305204, 11.63561100546868925214281016840, 12.58062642299700335337614887936, 13.91341388186580294138166018253

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