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Properties

Label 2-133-133.103-c1-0-1
Degree $2$
Conductor $133$
Sign $-0.272 - 0.962i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.395 − 0.228i)2-s − 1.79·3-s + (−0.895 + 1.55i)4-s + (−1.10 + 0.637i)5-s + (−0.708 + 0.409i)6-s + (0.5 + 2.59i)7-s + 1.73i·8-s + 0.208·9-s + (−0.291 + 0.504i)10-s + (1.5 + 2.59i)11-s + (1.60 − 2.77i)12-s + (−1.89 − 3.28i)13-s + (0.791 + 0.913i)14-s + (1.97 − 1.14i)15-s + (−1.39 − 2.41i)16-s + 4.37i·17-s + ⋯
L(s)  = 1  + (0.279 − 0.161i)2-s − 1.03·3-s + (−0.447 + 0.775i)4-s + (−0.493 + 0.285i)5-s + (−0.289 + 0.167i)6-s + (0.188 + 0.981i)7-s + 0.612i·8-s + 0.0695·9-s + (−0.0921 + 0.159i)10-s + (0.452 + 0.783i)11-s + (0.463 − 0.802i)12-s + (−0.525 − 0.910i)13-s + (0.211 + 0.244i)14-s + (0.510 − 0.294i)15-s + (−0.348 − 0.604i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.272 - 0.962i$
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: \((2,\ 133,\ (\ :1/2),\ -0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381339 + 0.504465i\)
\(L(\frac12)\) \(\approx\) \(0.381339 + 0.504465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 - 2.59i)T \)
19 \( 1 + (0.5 + 4.33i)T \)
good2 \( 1 + (-0.395 + 0.228i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 1.79T + 3T^{2} \)
5 \( 1 + (1.10 - 0.637i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.89 + 3.28i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.37iT - 17T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + (3.08 - 1.77i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.68 - 5.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.68 - 4.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.39 + 5.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.10iT - 47T^{2} \)
53 \( 1 + (-1.10 - 0.637i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 - 1.37iT - 61T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.08 + 1.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.20iT - 73T^{2} \)
79 \( 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 - 2.37T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.08301524634947966141167565705, −12.47402676586620192803748834531, −11.62329235189910255894424700187, −10.95511500227152373841382172862, −9.353247077701073814148860140897, −8.251965327038280284235404103329, −7.03782211199454810859535317659, −5.57029474243094228729055658868, −4.59685090398104915752461766496, −2.93202917874141033056948608756, 0.70724043955798438592555110953, 4.06768823508182958970816692935, 5.04397435011932766650442327025, 6.17734877036511453963306746055, 7.26819806612580722168801077797, 8.872486649907792761873170989116, 10.01982088799787724542870518649, 11.13119019317030110180235190971, 11.71513032870386300607400519522, 13.02790618402076707775479939862

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