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Properties

Label 2-140-140.107-c1-0-11
Degree $2$
Conductor $140$
Sign $0.999 + 0.0337i$
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  + (−0.599 + 1.28i)2-s + (0.147 − 0.551i)3-s + (−1.28 − 1.53i)4-s + (1.15 − 1.91i)5-s + (0.618 + 0.520i)6-s + (0.735 − 2.54i)7-s + (2.73 − 0.719i)8-s + (2.31 + 1.33i)9-s + (1.76 + 2.62i)10-s + (−3.42 + 1.98i)11-s + (−1.03 + 0.479i)12-s + (2.04 − 2.04i)13-s + (2.81 + 2.46i)14-s + (−0.888 − 0.918i)15-s + (−0.719 + 3.93i)16-s + (−0.155 + 0.580i)17-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (0.0853 − 0.318i)3-s + (−0.640 − 0.768i)4-s + (0.514 − 0.857i)5-s + (0.252 + 0.212i)6-s + (0.277 − 0.960i)7-s + (0.967 − 0.254i)8-s + (0.771 + 0.445i)9-s + (0.558 + 0.829i)10-s + (−1.03 + 0.597i)11-s + (−0.299 + 0.138i)12-s + (0.568 − 0.568i)13-s + (0.752 + 0.658i)14-s + (−0.229 − 0.237i)15-s + (−0.179 + 0.983i)16-s + (−0.0377 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0337i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.999 + 0.0337i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968397 - 0.0163474i\)
\(L(\frac12)\) \(\approx\) \(0.968397 - 0.0163474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 - 1.28i)T \)
5 \( 1 + (-1.15 + 1.91i)T \)
7 \( 1 + (-0.735 + 2.54i)T \)
good3 \( 1 + (-0.147 + 0.551i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (3.42 - 1.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.04 + 2.04i)T - 13iT^{2} \)
17 \( 1 + (0.155 - 0.580i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.00 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.39 + 0.640i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.25iT - 29T^{2} \)
31 \( 1 + (-2.83 + 1.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.29 - 2.49i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.35T + 41T^{2} \)
43 \( 1 + (3.31 + 3.31i)T + 43iT^{2} \)
47 \( 1 + (-3.24 - 12.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.90 - 0.779i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-7.41 - 12.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.11 - 3.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.67 + 0.984i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.86iT - 71T^{2} \)
73 \( 1 + (-9.44 - 2.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.45 + 5.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.46 + 5.46i)T + 83iT^{2} \)
89 \( 1 + (3.35 + 1.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.11 - 1.11i)T + 97iT^{2} \)
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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

−13.28998336205367192772561837611, −12.65061025530135155266625066523, −10.53998345301821139550807114992, −10.14528994425466790490075247607, −8.695823386111270160116814592103, −7.82001654463975959017747291429, −6.91117484020116057726213277922, −5.40973049162239723873879057885, −4.46553662604100648614585640000, −1.40957235883127094120216417481, 2.19986676452856643319575936019, 3.50038730831371471685755954202, 5.17567608770292908630806188847, 6.75364512710845152193832980112, 8.244616616879961622500618264252, 9.241219375763108350920108848684, 10.16721418974019288768586672594, 11.01495139186743441057315114033, 11.92020321679702874369522220796, 13.12689671439597835864431586439

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