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Properties

Label 2-133-133.122-c1-0-11
Degree $2$
Conductor $133$
Sign $-0.870 - 0.492i$
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  − 2.69i·2-s + (−0.838 − 1.45i)3-s − 5.24·4-s − 0.126i·5-s + (−3.90 + 2.25i)6-s + (2.64 + 0.0757i)7-s + 8.72i·8-s + (0.0951 − 0.164i)9-s − 0.340·10-s + (−0.136 + 0.235i)11-s + (4.39 + 7.60i)12-s + (−2.09 − 3.62i)13-s + (0.203 − 7.11i)14-s + (−0.183 + 0.106i)15-s + 12.9·16-s + (−3.46 + 1.99i)17-s + ⋯
L(s)  = 1  − 1.90i·2-s + (−0.483 − 0.838i)3-s − 2.62·4-s − 0.0566i·5-s + (−1.59 + 0.920i)6-s + (0.999 + 0.0286i)7-s + 3.08i·8-s + (0.0317 − 0.0549i)9-s − 0.107·10-s + (−0.0410 + 0.0710i)11-s + (1.26 + 2.19i)12-s + (−0.580 − 1.00i)13-s + (0.0544 − 1.90i)14-s + (−0.0474 + 0.0274i)15-s + 3.24·16-s + (−0.839 + 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.870 - 0.492i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208966 + 0.794488i\)
\(L(\frac12)\) \(\approx\) \(0.208966 + 0.794488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.64 - 0.0757i)T \)
19 \( 1 + (-2.84 + 3.30i)T \)
good2 \( 1 + 2.69iT - 2T^{2} \)
3 \( 1 + (0.838 + 1.45i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.126iT - 5T^{2} \)
11 \( 1 + (0.136 - 0.235i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.09 + 3.62i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.46 - 1.99i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.95 - 5.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.55 + 3.78i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.804 - 1.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.34 + 4.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.22 + 3.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.387 + 0.671i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.36 - 0.788i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.52iT - 53T^{2} \)
59 \( 1 + (-2.44 - 4.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.139 - 0.0803i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 - 6.69iT - 67T^{2} \)
71 \( 1 + (5.63 + 3.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.8 - 7.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 - 17.7iT - 83T^{2} \)
89 \( 1 + (-0.994 + 1.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.94 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.47153879056057070810980372573, −11.66335331320755264066311905857, −10.97270483413466346866296734413, −9.912252167405243323408859120114, −8.721291437633744444049125867828, −7.53378858676978011630152126428, −5.52785047949076141369320944418, −4.30726298965050398386257727456, −2.52304126107293553575527459913, −1.03044080881971883813807441938, 4.50259163367319888076074449717, 4.83209671794513472337934621356, 6.20254200592805180147275817181, 7.34127681868482353385975188764, 8.372133864596924215988346842063, 9.406672204055314725304530210757, 10.48354435320460335665055528235, 11.83925106203366644348020933897, 13.35149469281486463914989313040, 14.41178382412885089830439910227

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