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Properties

Label 2-140-35.9-c5-0-5
Degree $2$
Conductor $140$
Sign $0.504 - 0.863i$
Analytic cond. $22.4537$
Root an. cond. $4.73853$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−26.3 + 15.2i)3-s + (−55.1 + 9.42i)5-s + (−93.8 − 89.4i)7-s + (341. − 590. i)9-s + (−147. − 255. i)11-s − 360. i·13-s + (1.30e3 − 1.08e3i)15-s + (−1.27e3 + 735. i)17-s + (−1.01e3 + 1.76e3i)19-s + (3.83e3 + 929. i)21-s + (−3.50e3 − 2.02e3i)23-s + (2.94e3 − 1.03e3i)25-s + 1.33e4i·27-s − 562.·29-s + (−3.18e3 − 5.51e3i)31-s + ⋯
L(s)  = 1  + (−1.68 + 0.975i)3-s + (−0.985 + 0.168i)5-s + (−0.723 − 0.690i)7-s + (1.40 − 2.43i)9-s + (−0.368 − 0.637i)11-s − 0.591i·13-s + (1.50 − 1.24i)15-s + (−1.06 + 0.617i)17-s + (−0.646 + 1.11i)19-s + (1.89 + 0.459i)21-s + (−1.38 − 0.796i)23-s + (0.943 − 0.332i)25-s + 3.52i·27-s − 0.124·29-s + (−0.594 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.504 - 0.863i$
Analytic conductor: \(22.4537\)
Root analytic conductor: \(4.73853\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :5/2),\ 0.504 - 0.863i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2341066996\)
\(L(\frac12)\) \(\approx\) \(0.2341066996\)
\(L(\frac{7}{2})\) 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 \( 1 + (55.1 - 9.42i)T \)
7 \( 1 + (93.8 + 89.4i)T \)
good3 \( 1 + (26.3 - 15.2i)T + (121.5 - 210. i)T^{2} \)
11 \( 1 + (147. + 255. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 360. iT - 3.71e5T^{2} \)
17 \( 1 + (1.27e3 - 735. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.01e3 - 1.76e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (3.50e3 + 2.02e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 562.T + 2.05e7T^{2} \)
31 \( 1 + (3.18e3 + 5.51e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.11e3 + 1.21e3i)T + (3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 502.T + 1.15e8T^{2} \)
43 \( 1 - 2.24e4iT - 1.47e8T^{2} \)
47 \( 1 + (-872. - 503. i)T + (1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.76e4 + 1.01e4i)T + (2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (7.72e3 + 1.33e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.17e4 + 2.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.72e3 + 1.57e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.60e4T + 1.80e9T^{2} \)
73 \( 1 + (-5.35e4 + 3.09e4i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.37e4 - 5.84e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 5.12e3iT - 3.93e9T^{2} \)
89 \( 1 + (4.06e4 - 7.03e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 9.37e4iT - 8.58e9T^{2} \)
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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

−12.24649343104546624168573536985, −11.09427388999331731213496056536, −10.65590852469161656177161921781, −9.764836034277049869871942922409, −8.122265152231966247873266226747, −6.63016507453804764177573089454, −5.86614158899714229364633084492, −4.35469584137982262087454276799, −3.69233060685902517355492506007, −0.43631688512695948514409958453, 0.25709337534498091958380291825, 2.09964612935678697562778469517, 4.42464849113646548969831525830, 5.50283349850871945593854390680, 6.76196943387701608166874106669, 7.30289892779008985794939505597, 8.820539993108385049789450472720, 10.38215664180576578372673073410, 11.43377937032587126368937399474, 12.00669681143007221830694083807

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