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Properties

Label 2-140-35.12-c1-0-1
Degree $2$
Conductor $140$
Sign $0.323 - 0.946i$
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  + (−0.221 + 0.827i)3-s + (−1.60 + 1.55i)5-s + (1.42 + 2.22i)7-s + (1.96 + 1.13i)9-s + (1.59 + 2.75i)11-s + (−4.30 − 4.30i)13-s + (−0.930 − 1.67i)15-s + (0.267 + 0.0717i)17-s + (2.95 − 5.11i)19-s + (−2.16 + 0.686i)21-s + (1.51 + 5.64i)23-s + (0.163 − 4.99i)25-s + (−3.19 + 3.19i)27-s − 9.49i·29-s + (3.16 − 1.82i)31-s + ⋯
L(s)  = 1  + (−0.128 + 0.477i)3-s + (−0.718 + 0.695i)5-s + (0.539 + 0.842i)7-s + (0.654 + 0.377i)9-s + (0.479 + 0.830i)11-s + (−1.19 − 1.19i)13-s + (−0.240 − 0.432i)15-s + (0.0649 + 0.0173i)17-s + (0.677 − 1.17i)19-s + (−0.471 + 0.149i)21-s + (0.315 + 1.17i)23-s + (0.0327 − 0.999i)25-s + (−0.613 + 0.613i)27-s − 1.76i·29-s + (0.567 − 0.327i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819773 + 0.586266i\)
\(L(\frac12)\) \(\approx\) \(0.819773 + 0.586266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 1.55i)T \)
7 \( 1 + (-1.42 - 2.22i)T \)
good3 \( 1 + (0.221 - 0.827i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.30 + 4.30i)T + 13iT^{2} \)
17 \( 1 + (-0.267 - 0.0717i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.95 + 5.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.51 - 5.64i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 9.49iT - 29T^{2} \)
31 \( 1 + (-3.16 + 1.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.653 - 0.175i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.09iT - 41T^{2} \)
43 \( 1 + (-4.70 + 4.70i)T - 43iT^{2} \)
47 \( 1 + (0.118 + 0.443i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.96 + 0.526i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.29 - 7.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.23 - 4.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0180 + 0.0674i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.14T + 71T^{2} \)
73 \( 1 + (-0.446 + 1.66i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.31 + 3.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + 83iT^{2} \)
89 \( 1 + (-1.19 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.65 + 5.65i)T - 97iT^{2} \)
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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

−13.30057594362835121083370646262, −12.08451749588257973845916081892, −11.44587339697207163735392897830, −10.25459332821783486960144946904, −9.444371283219081786415363983895, −7.86317285141959704176227673507, −7.16812269098289044909251602063, −5.39868007962468043690575674805, −4.33274928660251847485286996969, −2.62168941232299601794351779750, 1.23815652034621334078932202121, 3.84382899481305712348472687809, 4.89440575677926695590866883602, 6.70506579902386931212380830710, 7.54028359628198997774656859721, 8.645827958273667584693493770025, 9.834190369704800375607060265143, 11.16112157988023460988015320309, 12.08121111088494129571132012946, 12.70975450060064940825740105366

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