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Properties

Label 2-133-133.102-c1-0-11
Degree $2$
Conductor $133$
Sign $-0.995 + 0.0942i$
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  + (1.36 − 2.35i)2-s − 2.49·3-s + (−2.70 − 4.69i)4-s + (−0.614 + 1.06i)5-s + (−3.39 + 5.87i)6-s + (2.19 − 1.47i)7-s − 9.31·8-s + 3.20·9-s + (1.67 + 2.90i)10-s + (1.11 − 1.93i)11-s + (6.74 + 11.6i)12-s + (1.07 − 1.87i)13-s + (−0.473 − 7.19i)14-s + (1.53 − 2.65i)15-s + (−7.26 + 12.5i)16-s − 1.40·17-s + ⋯
L(s)  = 1  + (0.962 − 1.66i)2-s − 1.43·3-s + (−1.35 − 2.34i)4-s + (−0.275 + 0.476i)5-s + (−1.38 + 2.39i)6-s + (0.831 − 0.555i)7-s − 3.29·8-s + 1.06·9-s + (0.529 + 0.917i)10-s + (0.336 − 0.582i)11-s + (1.94 + 3.37i)12-s + (0.299 − 0.518i)13-s + (−0.126 − 1.92i)14-s + (0.395 − 0.685i)15-s + (−1.81 + 3.14i)16-s − 0.341·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.995 + 0.0942i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 133,\ (\ :1/2),\ -0.995 + 0.0942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0477340 - 1.01025i\)
\(L(\frac12)\) \(\approx\) \(0.0477340 - 1.01025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.19 + 1.47i)T \)
19 \( 1 + (-0.945 + 4.25i)T \)
good2 \( 1 + (-1.36 + 2.35i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 2.49T + 3T^{2} \)
5 \( 1 + (0.614 - 1.06i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.11 + 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.07 + 1.87i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + (3.53 - 6.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.00 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.43 + 2.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.18 - 7.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.55 + 2.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.229T + 47T^{2} \)
53 \( 1 + (-0.119 - 0.206i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.41T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + (-1.17 - 2.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.11 - 1.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + (-2.72 + 4.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.62T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (3.50 + 6.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.56589593422386675412822346158, −11.36316315258936666771890035187, −11.10455341815764597007500794870, −10.60375381011559693558192221715, −9.065768002898756105519121383628, −6.84732309029497222952788208089, −5.46644200397411931042535978720, −4.71141537887261885425137520158, −3.28253666540408573829943754602, −1.04606318169455565671731039545, 4.25482685605920005415911440362, 5.04601178863264313124655881315, 5.92171212170493456124616158330, 6.90359614405767891928565492413, 8.054170254915382831664578864337, 9.162236533752142782482939433377, 11.21751440108749449650428724769, 12.10901587486980742398040092252, 12.66676961002971980519923588569, 13.92768731007328540206802670574

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