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Properties

Label 2-140-35.4-c1-0-2
Degree $2$
Conductor $140$
Sign $0.996 - 0.0828i$
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  + (1.5 + 0.866i)3-s + (−0.5 − 2.17i)5-s + (2.63 + 0.209i)7-s + (−1.13 + 1.97i)11-s + 6.09i·13-s + (1.13 − 3.70i)15-s + (−4.13 − 2.38i)17-s + (−2.13 − 3.70i)19-s + (3.77 + 2.59i)21-s + (−0.774 + 0.447i)23-s + (−4.50 + 2.17i)25-s − 5.19i·27-s + 3.27·29-s + (2.13 − 3.70i)31-s + (−3.41 + 1.97i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.223 − 0.974i)5-s + (0.996 + 0.0791i)7-s + (−0.342 + 0.594i)11-s + 1.68i·13-s + (0.293 − 0.955i)15-s + (−1.00 − 0.579i)17-s + (−0.490 − 0.849i)19-s + (0.823 + 0.566i)21-s + (−0.161 + 0.0932i)23-s + (−0.900 + 0.435i)25-s − 0.999i·27-s + 0.608·29-s + (0.383 − 0.664i)31-s + (−0.594 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.996 - 0.0828i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.996 - 0.0828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38302 + 0.0573886i\)
\(L(\frac12)\) \(\approx\) \(1.38302 + 0.0573886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.17i)T \)
7 \( 1 + (-2.63 - 0.209i)T \)
good3 \( 1 + (-1.5 - 0.866i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.09iT - 13T^{2} \)
17 \( 1 + (4.13 + 2.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 + 3.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.774 - 0.447i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + (-2.13 + 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.86 - 2.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 6.50iT - 43T^{2} \)
47 \( 1 + (-1.86 + 1.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.13 + 3.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.774 + 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.0 - 6.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-1.86 - 1.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.137 + 0.238i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−13.41489017695208431294110186176, −12.02328787062225009370653012714, −11.33987220296309440094317180332, −9.755564704351381770136614030483, −8.891019769530485354095453845206, −8.297116144780486952166476848643, −6.85207858241255026999163430946, −4.87039846142984731138562535907, −4.21641432278640671822632457388, −2.15059284060156860198596146756, 2.25357071367651651651678217180, 3.53518049086950142524561968582, 5.42045278587186085757282105169, 6.91060570836426987869113010442, 8.200790875209265107201527491740, 8.320085466032543741524198492635, 10.43020009454827993098495440921, 10.85060706658071379834919172213, 12.21891497807597239435482953177, 13.41472960446207745105320264048

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