Properties

Label 2-133-133.103-c1-0-2
Degree $2$
Conductor $133$
Sign $0.185 - 0.982i$
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  + (−2.33 + 1.34i)2-s − 1.67·3-s + (2.62 − 4.53i)4-s + (0.109 − 0.0633i)5-s + (3.90 − 2.25i)6-s + (2.64 − 0.0757i)7-s + 8.72i·8-s − 0.190·9-s + (−0.170 + 0.295i)10-s + (−0.136 − 0.235i)11-s + (−4.39 + 7.60i)12-s + (2.09 + 3.62i)13-s + (−6.06 + 3.73i)14-s + (−0.183 + 0.106i)15-s + (−6.49 − 11.2i)16-s + 3.99i·17-s + ⋯
L(s)  = 1  + (−1.64 + 0.951i)2-s − 0.967·3-s + (1.31 − 2.26i)4-s + (0.0490 − 0.0283i)5-s + (1.59 − 0.920i)6-s + (0.999 − 0.0286i)7-s + 3.08i·8-s − 0.0634·9-s + (−0.0538 + 0.0933i)10-s + (−0.0410 − 0.0710i)11-s + (−1.26 + 2.19i)12-s + (0.580 + 1.00i)13-s + (−1.61 + 0.998i)14-s + (−0.0474 + 0.0274i)15-s + (−1.62 − 2.81i)16-s + 0.969i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.185 - 0.982i$
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: \((2,\ 133,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317548 + 0.263098i\)
\(L(\frac12)\) \(\approx\) \(0.317548 + 0.263098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.28 - 0.809i)T \)
good2 \( 1 + (2.33 - 1.34i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + (-0.109 + 0.0633i)T + (2.5 - 4.33i)T^{2} \)
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 \( 1 - 3.99iT - 17T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
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 \( 1 - 1.57iT - 47T^{2} \)
53 \( 1 + (3.05 + 1.76i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 0.160iT - 61T^{2} \)
67 \( 1 + (5.79 + 3.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.63 + 3.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.8iT - 73T^{2} \)
79 \( 1 + (-9.55 + 5.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.7iT - 83T^{2} \)
89 \( 1 - 1.98T + 89T^{2} \)
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

−13.89333109873978121436559375117, −11.82227133375511059227708966725, −11.15382444437751769955079667666, −10.37876101693909282966159748635, −9.074067588407739688866386138479, −8.262814284552788498265747424924, −7.09935736148386126185905755275, −6.07789742571317408746933203921, −5.09667344261770166251074971552, −1.40178887540260312998881133733, 0.971869357788667737543888609413, 2.94491650537500803608065254650, 5.15859243255507637157226388272, 6.88916907992399609031323645350, 8.045456577531209016717320329963, 8.906417828917398405201466067892, 10.23829976449069859409176062706, 10.89643080048096400370441680918, 11.68679222244357861355393813577, 12.28035855790583498362610220273

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