↑

Properties

Label 2-133-133.103-c1-0-10
Degree $2$
Conductor $133$
Sign $0.621 + 0.783i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
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  + (2.11 − 1.21i)2-s + 0.519·3-s + (1.96 − 3.40i)4-s + (−2.56 + 1.47i)5-s + (1.09 − 0.632i)6-s + (0.877 + 2.49i)7-s − 4.71i·8-s − 2.73·9-s + (−3.60 + 6.24i)10-s + (−2.13 − 3.69i)11-s + (1.02 − 1.77i)12-s + (0.490 + 0.848i)13-s + (4.89 + 4.19i)14-s + (−1.33 + 0.768i)15-s + (−1.81 − 3.13i)16-s − 4.99i·17-s + ⋯
L(s)  = 1  + (1.49 − 0.861i)2-s + 0.299·3-s + (0.984 − 1.70i)4-s + (−1.14 + 0.661i)5-s + (0.447 − 0.258i)6-s + (0.331 + 0.943i)7-s − 1.66i·8-s − 0.910·9-s + (−1.13 + 1.97i)10-s + (−0.642 − 1.11i)11-s + (0.295 − 0.511i)12-s + (0.135 + 0.235i)13-s + (1.30 + 1.12i)14-s + (−0.343 + 0.198i)15-s + (−0.452 − 0.784i)16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.621 + 0.783i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83748 - 0.888392i\)
\(L(\frac12)\) \(\approx\) \(1.83748 - 0.888392i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.877 - 2.49i)T \)
19 \( 1 + (-1.53 - 4.07i)T \)
good2 \( 1 + (-2.11 + 1.21i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 0.519T + 3T^{2} \)
5 \( 1 + (2.56 - 1.47i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.13 + 3.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.490 - 0.848i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.99iT - 17T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 + (-1.16 + 0.674i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.73 + 4.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.05 + 0.611i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.34 - 2.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.72 - 4.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.39iT - 47T^{2} \)
53 \( 1 + (-6.59 - 3.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.34T + 59T^{2} \)
61 \( 1 - 3.37iT - 61T^{2} \)
67 \( 1 + (1.50 + 0.869i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.4 + 7.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.63iT - 73T^{2} \)
79 \( 1 + (4.21 - 2.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.39iT - 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (-7.58 + 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.16279738709227281188563298161, −11.77532108680217558746941161353, −11.57123719974851647880254276002, −10.72376649981835694893469364297, −8.920453012703103530070283315669, −7.70680118560682091892734244170, −6.01353764380281197249247243635, −5.01702300190554006196795664577, −3.39181981719551954086264329414, −2.75980639365218901105545116466, 3.29086864142919688578784963342, 4.44021427449458626676963438529, 5.23796882008971385966191194060, 6.96505531628453294274001017474, 7.71899527203791971353909172222, 8.674175227410196203293061683483, 10.71431735255911778651490372987, 11.80219744449588043803193458302, 12.76018169870003352725898599517, 13.39090690322505918111048333238

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