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Properties

Label 2-140-140.123-c1-0-0
Degree $2$
Conductor $140$
Sign $-0.878 - 0.478i$
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  + (0.157 + 1.40i)2-s + (0.290 + 1.08i)3-s + (−1.95 + 0.442i)4-s + (−2.11 + 0.727i)5-s + (−1.47 + 0.578i)6-s + (−2.16 + 1.51i)7-s + (−0.928 − 2.67i)8-s + (1.50 − 0.871i)9-s + (−1.35 − 2.85i)10-s + (2.58 + 1.49i)11-s + (−1.04 − 1.98i)12-s + (4.05 + 4.05i)13-s + (−2.47 − 2.80i)14-s + (−1.40 − 2.07i)15-s + (3.60 − 1.72i)16-s + (0.617 + 2.30i)17-s + ⋯
L(s)  = 1  + (0.111 + 0.993i)2-s + (0.167 + 0.625i)3-s + (−0.975 + 0.221i)4-s + (−0.945 + 0.325i)5-s + (−0.602 + 0.236i)6-s + (−0.819 + 0.573i)7-s + (−0.328 − 0.944i)8-s + (0.503 − 0.290i)9-s + (−0.428 − 0.903i)10-s + (0.779 + 0.449i)11-s + (−0.301 − 0.572i)12-s + (1.12 + 1.12i)13-s + (−0.661 − 0.750i)14-s + (−0.361 − 0.536i)15-s + (0.902 − 0.431i)16-s + (0.149 + 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.878 - 0.478i$
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: \((2,\ 140,\ (\ :1/2),\ -0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228349 + 0.896031i\)
\(L(\frac12)\) \(\approx\) \(0.228349 + 0.896031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.157 - 1.40i)T \)
5 \( 1 + (2.11 - 0.727i)T \)
7 \( 1 + (2.16 - 1.51i)T \)
good3 \( 1 + (-0.290 - 1.08i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.58 - 1.49i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.05 - 4.05i)T + 13iT^{2} \)
17 \( 1 + (-0.617 - 2.30i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.25 + 2.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.79 + 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (-3.12 - 1.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.61 - 1.23i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (-7.62 + 7.62i)T - 43iT^{2} \)
47 \( 1 + (-0.765 + 2.85i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.47 - 0.662i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.04 + 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.950 - 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.00 + 0.805i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.09iT - 71T^{2} \)
73 \( 1 + (9.41 - 2.52i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.03 - 6.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.99 - 5.99i)T - 83iT^{2} \)
89 \( 1 + (1.77 - 1.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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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

−13.86110647974966367739879563245, −12.61142913658274935359207545223, −11.78043664137726709127326619783, −10.21976253059368060500771212114, −9.208326027924994459212120954574, −8.431075385808702164619616288465, −6.92797124075703220039690241827, −6.25010270752784839421145575320, −4.32394567049861705212896606549, −3.69744815068364833388565949659, 1.02184339697857863559871320338, 3.28666559178248426858302987056, 4.25509046600697140468500728670, 6.09036597421893536970267978766, 7.64598622193830144702175679446, 8.513803210512031202013524474195, 9.829478630396090508591766702159, 10.82889159379588126649770833046, 11.85268264156491495907322766406, 12.75932646313198705214717695990

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