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Properties

Label 2-133-133.103-c1-0-4
Degree $2$
Conductor $133$
Sign $0.610 + 0.792i$
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  + (−0.795 + 0.459i)2-s − 3.24·3-s + (−0.578 + 1.00i)4-s + (1.25 − 0.724i)5-s + (2.58 − 1.49i)6-s + (0.309 − 2.62i)7-s − 2.89i·8-s + 7.55·9-s + (−0.665 + 1.15i)10-s + (0.0421 + 0.0730i)11-s + (1.87 − 3.25i)12-s + (0.416 + 0.720i)13-s + (0.959 + 2.23i)14-s + (−4.07 + 2.35i)15-s + (0.173 + 0.300i)16-s − 7.66i·17-s + ⋯
L(s)  = 1  + (−0.562 + 0.324i)2-s − 1.87·3-s + (−0.289 + 0.501i)4-s + (0.561 − 0.324i)5-s + (1.05 − 0.608i)6-s + (0.117 − 0.993i)7-s − 1.02i·8-s + 2.51·9-s + (−0.210 + 0.364i)10-s + (0.0127 + 0.0220i)11-s + (0.542 − 0.939i)12-s + (0.115 + 0.199i)13-s + (0.256 + 0.596i)14-s + (−1.05 + 0.607i)15-s + (0.0433 + 0.0751i)16-s − 1.85i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.610 + 0.792i$
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: \((2,\ 133,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357898 - 0.176097i\)
\(L(\frac12)\) \(\approx\) \(0.357898 - 0.176097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 2.62i)T \)
19 \( 1 + (3.13 + 3.02i)T \)
good2 \( 1 + (0.795 - 0.459i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 3.24T + 3T^{2} \)
5 \( 1 + (-1.25 + 0.724i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0421 - 0.0730i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.416 - 0.720i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.66iT - 17T^{2} \)
23 \( 1 - 3.30T + 23T^{2} \)
29 \( 1 + (-2.70 + 1.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.122 + 0.213i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.96 + 1.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.33 + 7.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.43 + 2.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.85iT - 47T^{2} \)
53 \( 1 + (2.45 + 1.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.10T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + (6.07 + 3.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.408 - 0.235i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + (-4.61 + 2.66i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.16iT - 83T^{2} \)
89 \( 1 + 5.24T + 89T^{2} \)
97 \( 1 + (-3.51 + 6.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.03064470392288966644100499896, −11.98824751475902445157709152798, −11.05055378717393689493245992643, −10.06576067261899310910460635999, −9.126510785663265134329293768437, −7.32475063414870242669467443548, −6.77849428660726436731329974606, −5.28690476927452571149797202063, −4.28315806612249369774309725124, −0.66710323658940246562214893951, 1.66305149695422660755233179576, 4.71042849322654552550437243857, 5.91493727059478947755081261263, 6.24528521704499946556715226908, 8.298394769718788894004204539426, 9.679922947003243294264177234133, 10.52801372468037373421760098313, 11.06786164951773705838653058688, 12.20435940487831354185644159139, 12.97231312766999632761923769913

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