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Properties

Label 2-140-28.3-c1-0-4
Degree $2$
Conductor $140$
Sign $-0.926 - 0.375i$
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  + (0.569 + 1.29i)2-s + (−1.51 + 2.62i)3-s + (−1.35 + 1.47i)4-s + (0.866 − 0.5i)5-s + (−4.25 − 0.465i)6-s + (2.57 − 0.602i)7-s + (−2.67 − 0.908i)8-s + (−3.08 − 5.33i)9-s + (1.14 + 0.836i)10-s + (−1.03 − 0.598i)11-s + (−1.82 − 5.77i)12-s + 4.83i·13-s + (2.24 + 2.99i)14-s + 3.02i·15-s + (−0.349 − 3.98i)16-s + (2.20 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.402 + 0.915i)2-s + (−0.873 + 1.51i)3-s + (−0.675 + 0.737i)4-s + (0.387 − 0.223i)5-s + (−1.73 − 0.190i)6-s + (0.973 − 0.227i)7-s + (−0.946 − 0.321i)8-s + (−1.02 − 1.77i)9-s + (0.360 + 0.264i)10-s + (−0.312 − 0.180i)11-s + (−0.525 − 1.66i)12-s + 1.34i·13-s + (0.600 + 0.799i)14-s + 0.781i·15-s + (−0.0873 − 0.996i)16-s + (0.534 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.926 - 0.375i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198836 + 1.02001i\)
\(L(\frac12)\) \(\approx\) \(0.198836 + 1.02001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.569 - 1.29i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.57 + 0.602i)T \)
good3 \( 1 + (1.51 - 2.62i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.03 + 0.598i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.83iT - 13T^{2} \)
17 \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.711 - 1.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.02 + 2.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.774T + 29T^{2} \)
31 \( 1 + (3.31 - 5.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.55 + 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.46iT - 41T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + (0.535 + 0.927i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.68 - 2.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.94 + 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.31 + 4.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.14 - 5.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 16.3iT - 71T^{2} \)
73 \( 1 + (-0.0927 - 0.0535i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.32 - 5.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + (3.41 - 1.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.71iT - 97T^{2} \)
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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

−14.11953565772674420278914059968, −12.54851804716458512110823936455, −11.50031179172580300846309057329, −10.56220649489871240342678202102, −9.387100505836601814014068073049, −8.521438527332748254797914360043, −6.88040243928113664705493925265, −5.53971093279513854780994884340, −4.87067121264464227233600269064, −3.84563382771624265125241478278, 1.20843076145750671849511129668, 2.66005581652624958284829628240, 5.15182826892105798230897534499, 5.75579027217877158460264654834, 7.25564180615425338757464373860, 8.352482191214227891992541769462, 10.04227909385836316686108685519, 11.16614773789455902606291871770, 11.65842055369998038053049962897, 12.80459445970512416008196276157

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