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Properties

Label 2-1134-21.5-c1-0-13
Degree $2$
Conductor $1134$
Sign $0.0864 - 0.996i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.529 + 0.917i)5-s + (1.22 − 2.34i)7-s + 0.999i·8-s + (−0.917 + 0.529i)10-s + (−3.99 + 2.30i)11-s + 0.109i·13-s + (2.23 − 1.41i)14-s + (−0.5 + 0.866i)16-s + (2.07 + 3.59i)17-s + (6.05 + 3.49i)19-s − 1.05·20-s − 4.61·22-s + (5.36 + 3.09i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.236 + 0.410i)5-s + (0.464 − 0.885i)7-s + 0.353i·8-s + (−0.290 + 0.167i)10-s + (−1.20 + 0.696i)11-s + 0.0304i·13-s + (0.597 − 0.377i)14-s + (−0.125 + 0.216i)16-s + (0.503 + 0.872i)17-s + (1.38 + 0.801i)19-s − 0.236·20-s − 0.984·22-s + (1.11 + 0.645i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0864 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.0864 - 0.996i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.0864 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.155153355\)
\(L(\frac12)\) \(\approx\) \(2.155153355\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.22 + 2.34i)T \)
good5 \( 1 + (0.529 - 0.917i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.99 - 2.30i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.109iT - 13T^{2} \)
17 \( 1 + (-2.07 - 3.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.05 - 3.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.36 - 3.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.90iT - 29T^{2} \)
31 \( 1 + (7.12 - 4.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.62 + 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.183T + 41T^{2} \)
43 \( 1 - 3.94T + 43T^{2} \)
47 \( 1 + (2.08 - 3.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.89 + 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.09 + 8.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.17 + 4.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0671 - 0.116i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-12.4 + 7.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.988 + 1.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 + (-0.355 + 0.615i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.72iT - 97T^{2} \)
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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

−10.20598691389135309636564181045, −9.194743602233624719713197533998, −7.85108832505251148534926898872, −7.57394975154306837191051123757, −6.85912739693887284095735635280, −5.52080793137083812649748607567, −5.02151822908516322415207508791, −3.81899205655868884564219274598, −3.10134222689762278678122846506, −1.55757593126820406771980635459, 0.819423468251759532262625173852, 2.52100777279386910843037361612, 3.12089484337705140897191725920, 4.62901359652437056281229310943, 5.22801141167028043606355549542, 5.86708605470280818614879796608, 7.19007287497997870764057732835, 7.972815430344203350086897859597, 8.883747758811328541726248348734, 9.575196085468856720885349535924

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