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Properties

Label 2-140-7.5-c2-0-3
Degree $2$
Conductor $140$
Sign $0.00344 + 0.999i$
Analytic cond. $3.81472$
Root an. cond. $1.95313$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.70 + 1.56i)3-s + (−1.93 − 1.11i)5-s + (2.68 − 6.46i)7-s + (0.367 − 0.636i)9-s + (−3.55 − 6.16i)11-s − 21.4i·13-s + 6.97·15-s + (29.3 − 16.9i)17-s + (−24.4 − 14.1i)19-s + (2.83 + 21.6i)21-s + (−15.1 + 26.2i)23-s + (2.5 + 4.33i)25-s − 25.7i·27-s + 10.1·29-s + (−42.2 + 24.3i)31-s + ⋯
L(s)  = 1  + (−0.900 + 0.520i)3-s + (−0.387 − 0.223i)5-s + (0.383 − 0.923i)7-s + (0.0408 − 0.0707i)9-s + (−0.323 − 0.560i)11-s − 1.65i·13-s + 0.465·15-s + (1.72 − 0.997i)17-s + (−1.28 − 0.742i)19-s + (0.135 + 1.03i)21-s + (−0.658 + 1.14i)23-s + (0.100 + 0.173i)25-s − 0.955i·27-s + 0.350·29-s + (−1.36 + 0.786i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00344 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00344 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.00344 + 0.999i$
Analytic conductor: \(3.81472\)
Root analytic conductor: \(1.95313\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1),\ 0.00344 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.511388 - 0.509629i\)
\(L(\frac12)\) \(\approx\) \(0.511388 - 0.509629i\)
\(L(2)\) 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 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-2.68 + 6.46i)T \)
good3 \( 1 + (2.70 - 1.56i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (3.55 + 6.16i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + (-29.3 + 16.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (24.4 + 14.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (15.1 - 26.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 10.1T + 841T^{2} \)
31 \( 1 + (42.2 - 24.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (4.57 - 7.92i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 8.85iT - 1.68e3T^{2} \)
43 \( 1 - 34.5T + 1.84e3T^{2} \)
47 \( 1 + (9.92 + 5.72i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (15.3 + 26.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.4 + 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-24.3 - 14.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.5 + 51.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 38.7T + 5.04e3T^{2} \)
73 \( 1 + (68.2 - 39.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-15.8 + 27.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 31.5iT - 6.88e3T^{2} \)
89 \( 1 + (-108. - 62.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 172. iT - 9.40e3T^{2} \)
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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

−12.56121387278919255059729684814, −11.42379819068764751443009638940, −10.70514300201907610836849847823, −9.955324529708152866376087267769, −8.256478868156978934391104791676, −7.42292658233289840058914056724, −5.69269833194685253671199291314, −4.92739229838127237206932206774, −3.42851287794906565437247131004, −0.53535479476420895180069883189, 1.93527869732373900589333486305, 4.12420231074025252704452456403, 5.66478512723179582609574799209, 6.49712238907119939095932617404, 7.77907840211061739537646596701, 8.913192236645179659781417509351, 10.31609175828799894957984075434, 11.38367947849326754427069273002, 12.26599773344839076681113656561, 12.58947028100346476701342449901

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