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Properties

Label 2-133-133.102-c1-0-6
Degree $2$
Conductor $133$
Sign $0.962 + 0.271i$
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  + (−1.16 + 2.01i)2-s − 1.27·3-s + (−1.69 − 2.93i)4-s + (2.05 − 3.55i)5-s + (1.48 − 2.56i)6-s + (−0.208 − 2.63i)7-s + 3.23·8-s − 1.36·9-s + (4.76 + 8.25i)10-s + (−1.36 + 2.36i)11-s + (2.16 + 3.75i)12-s + (2.05 − 3.55i)13-s + (5.54 + 2.64i)14-s + (−2.61 + 4.53i)15-s + (−0.360 + 0.624i)16-s + 1.24·17-s + ⋯
L(s)  = 1  + (−0.820 + 1.42i)2-s − 0.737·3-s + (−0.847 − 1.46i)4-s + (0.917 − 1.58i)5-s + (0.605 − 1.04i)6-s + (−0.0789 − 0.996i)7-s + 1.14·8-s − 0.456·9-s + (1.50 + 2.60i)10-s + (−0.412 + 0.713i)11-s + (0.625 + 1.08i)12-s + (0.569 − 0.986i)13-s + (1.48 + 0.706i)14-s + (−0.676 + 1.17i)15-s + (−0.0901 + 0.156i)16-s + 0.302·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.962 + 0.271i$
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: \((2,\ 133,\ (\ :1/2),\ 0.962 + 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529041 - 0.0732882i\)
\(L(\frac12)\) \(\approx\) \(0.529041 - 0.0732882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.208 + 2.63i)T \)
19 \( 1 + (4.34 + 0.339i)T \)
good2 \( 1 + (1.16 - 2.01i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 1.27T + 3T^{2} \)
5 \( 1 + (-2.05 + 3.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.05 + 3.55i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
23 \( 1 - 5.63T + 23T^{2} \)
29 \( 1 + (-0.386 + 0.669i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.21 - 2.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.32 + 2.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.41 - 4.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.250 + 0.434i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.00235T + 47T^{2} \)
53 \( 1 + (-0.280 - 0.485i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.61T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (4.70 + 8.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.28 - 9.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + (0.675 - 1.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 + (-1.72 - 2.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.24294988989198606206992908160, −12.53843400095987329680641795842, −10.75121288514408208183277255650, −9.836199898554764562327304919136, −8.804204364972950380621275941300, −7.970444161792871124326352868422, −6.60641119191435892258125879673, −5.58213036173515321544890304138, −4.85113713332117113547854454375, −0.802541341592220561337579294404, 2.21308869103481245661817628151, 3.21255923565107538142613056924, 5.70952394450989302493954391839, 6.59288062740105341781257303069, 8.537091325016231210903393437425, 9.442588985094819343456381296663, 10.59875838895512652372023038635, 11.06882070731908241986177455782, 11.75308655083556624102569435698, 12.97103781302439113454228674620

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