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Properties

Label 2-140-140.139-c1-0-12
Degree $2$
Conductor $140$
Sign $0.817 - 0.576i$
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  + (1.06 + 0.927i)2-s − 0.662i·3-s + (0.280 + 1.98i)4-s + (1.31 − 1.81i)5-s + (0.613 − 0.707i)6-s + (−1.19 + 2.35i)7-s + (−1.53 + 2.37i)8-s + 2.56·9-s + (3.07 − 0.718i)10-s − 3.09i·11-s + (1.31 − 0.185i)12-s − 4.66·13-s + (−3.46 + 1.40i)14-s + (−1.19 − 0.868i)15-s + (−3.84 + 1.11i)16-s + 2.04·17-s + ⋯
L(s)  = 1  + (0.755 + 0.655i)2-s − 0.382i·3-s + (0.140 + 0.990i)4-s + (0.586 − 0.810i)5-s + (0.250 − 0.288i)6-s + (−0.453 + 0.891i)7-s + (−0.543 + 0.839i)8-s + 0.853·9-s + (0.973 − 0.227i)10-s − 0.932i·11-s + (0.378 − 0.0536i)12-s − 1.29·13-s + (−0.926 + 0.375i)14-s + (−0.309 − 0.224i)15-s + (−0.960 + 0.277i)16-s + 0.496·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.817 - 0.576i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.817 - 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54382 + 0.489926i\)
\(L(\frac12)\) \(\approx\) \(1.54382 + 0.489926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.927i)T \)
5 \( 1 + (-1.31 + 1.81i)T \)
7 \( 1 + (1.19 - 2.35i)T \)
good3 \( 1 + 0.662iT - 3T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 4.66T + 13T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 + 1.87T + 23T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 - 0.290iT - 47T^{2} \)
53 \( 1 - 9.49iT - 53T^{2} \)
59 \( 1 - 8.05T + 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 - 2.39T + 67T^{2} \)
71 \( 1 - 9.65iT - 71T^{2} \)
73 \( 1 - 4.09T + 73T^{2} \)
79 \( 1 - 1.35iT - 79T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 + 2.82iT - 89T^{2} \)
97 \( 1 + 6.14T + 97T^{2} \)
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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

−13.09478013975788233921046218055, −12.60225437371725425303228849331, −11.80421494293494635996344210300, −9.992360319567107023714687497381, −8.870535821680348876454888372998, −7.86593019764070768660111124496, −6.48927843624590705435568627971, −5.63015221986789320831414535245, −4.40756131001684200895386632554, −2.48131089780307161443287760164, 2.21142003745176111209483557497, 3.81233587185757933520743069303, 4.88342088109106914634209227075, 6.49742684006542914731320931860, 7.29500060549339542709130291679, 9.764393984767971062318364401899, 9.985248572680016739659082522202, 10.82360330650690020876355185259, 12.25160026360757964665684759399, 13.01101278994078573921581108862

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