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Properties

Label 2-140-140.107-c1-0-9
Degree $2$
Conductor $140$
Sign $0.929 + 0.367i$
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  + (−1.15 − 0.809i)2-s + (−0.147 + 0.551i)3-s + (0.690 + 1.87i)4-s + (1.15 − 1.91i)5-s + (0.618 − 0.520i)6-s + (−0.735 + 2.54i)7-s + (0.719 − 2.73i)8-s + (2.31 + 1.33i)9-s + (−2.88 + 1.29i)10-s + (3.42 − 1.98i)11-s + (−1.13 + 0.103i)12-s + (2.04 − 2.04i)13-s + (2.90 − 2.35i)14-s + (0.888 + 0.918i)15-s + (−3.04 + 2.59i)16-s + (−0.155 + 0.580i)17-s + ⋯
L(s)  = 1  + (−0.820 − 0.572i)2-s + (−0.0853 + 0.318i)3-s + (0.345 + 0.938i)4-s + (0.514 − 0.857i)5-s + (0.252 − 0.212i)6-s + (−0.277 + 0.960i)7-s + (0.254 − 0.967i)8-s + (0.771 + 0.445i)9-s + (−0.912 + 0.408i)10-s + (1.03 − 0.597i)11-s + (−0.328 + 0.0297i)12-s + (0.568 − 0.568i)13-s + (0.777 − 0.628i)14-s + (0.229 + 0.237i)15-s + (−0.761 + 0.647i)16-s + (−0.0377 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.929 + 0.367i$
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: \((2,\ 140,\ (\ :1/2),\ 0.929 + 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835102 - 0.159080i\)
\(L(\frac12)\) \(\approx\) \(0.835102 - 0.159080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.809i)T \)
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

−12.80030736227591634776434416765, −12.08043361126402327491771099485, −10.93614647278149834652689481493, −9.875111183411218881510707681784, −9.023697558983891458871762523978, −8.338207348422287199667315929431, −6.67176918321184019834597616346, −5.24411745706308979057425411373, −3.56896428883748279684767027922, −1.64053576479233959535615754039, 1.62735153914142671557160614790, 4.03596012374073146411046609334, 6.14606825656098142790279732320, 6.82712679783044753393805615963, 7.59339276159690433967471730242, 9.292157504875942282602183970173, 9.948232506454421531735362185963, 10.88263109995918165512010883305, 12.04228959687087367104248981111, 13.58786427397432330314897129960

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