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Properties

Label 2-140-140.123-c1-0-1
Degree $2$
Conductor $140$
Sign $-0.757 - 0.652i$
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  + (−1.24 + 0.674i)2-s + (0.664 + 2.47i)3-s + (1.09 − 1.67i)4-s + (−1.93 + 1.11i)5-s + (−2.49 − 2.63i)6-s + (2.23 + 1.41i)7-s + (−0.224 + 2.81i)8-s + (−3.10 + 1.79i)9-s + (1.65 − 2.69i)10-s + (−1.59 − 0.921i)11-s + (4.88 + 1.58i)12-s + (−2.94 − 2.94i)13-s + (−3.73 − 0.243i)14-s + (−4.04 − 4.06i)15-s + (−1.62 − 3.65i)16-s + (0.795 + 2.96i)17-s + ⋯
L(s)  = 1  + (−0.878 + 0.476i)2-s + (0.383 + 1.43i)3-s + (0.545 − 0.838i)4-s + (−0.867 + 0.497i)5-s + (−1.01 − 1.07i)6-s + (0.846 + 0.533i)7-s + (−0.0793 + 0.996i)8-s + (−1.03 + 0.597i)9-s + (0.524 − 0.851i)10-s + (−0.481 − 0.277i)11-s + (1.40 + 0.458i)12-s + (−0.817 − 0.817i)13-s + (−0.997 − 0.0651i)14-s + (−1.04 − 1.05i)15-s + (−0.405 − 0.914i)16-s + (0.192 + 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.757 - 0.652i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257583 + 0.693635i\)
\(L(\frac12)\) \(\approx\) \(0.257583 + 0.693635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.674i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-2.23 - 1.41i)T \)
good3 \( 1 + (-0.664 - 2.47i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.59 + 0.921i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 + 2.94i)T + 13iT^{2} \)
17 \( 1 + (-0.795 - 2.96i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.44 - 0.654i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.35iT - 29T^{2} \)
31 \( 1 + (-3.78 - 2.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.86 - 1.03i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (1.02 - 1.02i)T - 43iT^{2} \)
47 \( 1 + (-0.210 + 0.785i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.61 - 0.699i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.11 - 3.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.19 - 0.856i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + (-1.97 + 0.529i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.95 + 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.227 - 0.227i)T - 83iT^{2} \)
89 \( 1 + (3.75 - 2.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.196 + 0.196i)T - 97iT^{2} \)
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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

−14.25458942551511139615783132990, −12.13187343345495706111342918186, −11.03949374632741909969709816737, −10.38339660586531348173701905720, −9.456019536374684332062516321822, −8.199211167169101403101808835018, −7.74627875758279740713675836861, −5.78895296645241593423677349423, −4.60814306739284099061656498341, −2.92873295805024079938535832826, 1.03470017744885340154249762695, 2.63134332628744025059161654929, 4.61958822485128790214198787874, 7.15933140058057904990379598847, 7.38907233434394295042461260825, 8.376109058735594500082530605539, 9.394275003437276357986648527612, 11.02947382620191717600305207352, 11.78547334541216270154610532747, 12.55315450230805686799528943820

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