Properties

Label 2-140-35.13-c1-0-2
Degree $2$
Conductor $140$
Sign $0.732 + 0.680i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.386 − 0.386i)3-s + (0.386 − 2.20i)5-s + (0.0564 − 2.64i)7-s + 2.70i·9-s + 1.70·11-s + (0.386 − 0.386i)13-s + (−0.701 − i)15-s + (4.79 + 4.79i)17-s − 5.95·19-s + (−0.999 − 1.04i)21-s + (−2.70 − 2.70i)23-s + (−4.70 − 1.70i)25-s + (2.20 + 2.20i)27-s + 5.70i·29-s + 8.03i·31-s + ⋯
L(s)  = 1  + (0.223 − 0.223i)3-s + (0.172 − 0.984i)5-s + (0.0213 − 0.999i)7-s + 0.900i·9-s + 0.513·11-s + (0.107 − 0.107i)13-s + (−0.181 − 0.258i)15-s + (1.16 + 1.16i)17-s − 1.36·19-s + (−0.218 − 0.227i)21-s + (−0.563 − 0.563i)23-s + (−0.940 − 0.340i)25-s + (0.423 + 0.423i)27-s + 1.05i·29-s + 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.732 + 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11820 - 0.439200i\)
\(L(\frac12)\) \(\approx\) \(1.11820 - 0.439200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.386 + 2.20i)T \)
7 \( 1 + (-0.0564 + 2.64i)T \)
good3 \( 1 + (-0.386 + 0.386i)T - 3iT^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 + (-0.386 + 0.386i)T - 13iT^{2} \)
17 \( 1 + (-4.79 - 4.79i)T + 17iT^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + (2.70 + 2.70i)T + 23iT^{2} \)
29 \( 1 - 5.70iT - 29T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 + (-2.70 + 2.70i)T - 37iT^{2} \)
41 \( 1 - 5.95iT - 41T^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 + (3.24 + 3.24i)T + 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + (1.81 - 1.81i)T - 73iT^{2} \)
79 \( 1 - 0.298iT - 79T^{2} \)
83 \( 1 + (-4.13 + 4.13i)T - 83iT^{2} \)
89 \( 1 - 2.08T + 89T^{2} \)
97 \( 1 + (1.15 + 1.15i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.97808523065608087900037753519, −12.35661003590704821775542323959, −10.82149553925697742962762965316, −10.08453402833322882498749882019, −8.615012312782798724416002490286, −7.957645079198399874360244823297, −6.56443585428303825431845638620, −5.09351526967129225756535371441, −3.87686229710258682444746161906, −1.58709043505043042219040390675, 2.53491712664680416632808907380, 3.87169433268866987595556109588, 5.76321829058394977449345617334, 6.64919724837737443045715225100, 8.054245818192029379269257630979, 9.349280407172115413030993697339, 9.967146181298193543174278459176, 11.47859958548846544570912897548, 12.00094328533272118484002333043, 13.37830322462114385018595126544

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