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Properties

Label 2-140-28.19-c1-0-0
Degree $2$
Conductor $140$
Sign $0.581 - 0.813i$
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  + (−0.950 − 1.04i)2-s + (1.36 + 2.37i)3-s + (−0.192 + 1.99i)4-s + (−0.866 − 0.5i)5-s + (1.18 − 3.68i)6-s + (−1.02 + 2.43i)7-s + (2.26 − 1.69i)8-s + (−2.24 + 3.89i)9-s + (0.299 + 1.38i)10-s + (0.0868 − 0.0501i)11-s + (−4.98 + 2.26i)12-s + 4.11i·13-s + (3.52 − 1.24i)14-s − 2.73i·15-s + (−3.92 − 0.766i)16-s + (4.67 − 2.69i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.740i)2-s + (0.790 + 1.36i)3-s + (−0.0962 + 0.995i)4-s + (−0.387 − 0.223i)5-s + (0.482 − 1.50i)6-s + (−0.387 + 0.922i)7-s + (0.801 − 0.597i)8-s + (−0.748 + 1.29i)9-s + (0.0948 + 0.437i)10-s + (0.0261 − 0.0151i)11-s + (−1.43 + 0.654i)12-s + 1.14i·13-s + (0.942 − 0.333i)14-s − 0.706i·15-s + (−0.981 − 0.191i)16-s + (1.13 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811902 + 0.417583i\)
\(L(\frac12)\) \(\approx\) \(0.811902 + 0.417583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.950 + 1.04i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.02 - 2.43i)T \)
good3 \( 1 + (-1.36 - 2.37i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.0868 + 0.0501i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.11iT - 13T^{2} \)
17 \( 1 + (-4.67 + 2.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.72 + 6.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.30 - 0.754i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + (2.72 + 4.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.519 - 0.899i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.99iT - 41T^{2} \)
43 \( 1 - 7.04iT - 43T^{2} \)
47 \( 1 + (-2.22 + 3.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.07 + 5.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.26 - 7.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.84 - 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0950 - 0.0548i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.73iT - 71T^{2} \)
73 \( 1 + (5.32 - 3.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.70 - 2.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.50T + 83T^{2} \)
89 \( 1 + (2.76 + 1.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−13.23864699211808863816331132100, −11.91067687873637677864992685590, −11.25548506342399271575982456900, −9.868545632285551380015790362176, −9.279477480484963795300501472314, −8.687883475502797620204997102192, −7.31952043981650950930447790732, −5.04651717941627633690640738016, −3.76829500450089196327090485369, −2.68364643403115004492391290084, 1.23131842623866793757342007767, 3.39029276112199141427952699919, 5.74774595628150616684199815382, 6.96763827943764723412652806786, 7.74017069092279978508219660643, 8.247527785713931355364318451174, 9.762919244788860801803147489338, 10.68888175659962063833672196464, 12.29366445241778485217988935031, 13.16016231485467041353614730093

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