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Properties

Label 2-1134-63.25-c1-0-1
Degree $2$
Conductor $1134$
Sign $0.0477 - 0.998i$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (−2.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s + (2.5 + 4.33i)11-s + (2.5 + 0.866i)14-s + 16-s + (−2 + 3.46i)17-s + (−4 − 6.92i)19-s + (0.5 − 0.866i)20-s + (−2.5 − 4.33i)22-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + (0.753 + 1.30i)11-s + (0.668 + 0.231i)14-s + 0.250·16-s + (−0.485 + 0.840i)17-s + (−0.917 − 1.58i)19-s + (0.111 − 0.193i)20-s + (−0.533 − 0.923i)22-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.0477 - 0.998i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.0477 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7373549650\)
\(L(\frac12)\) \(\approx\) \(0.7373549650\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + (3.5 - 6.06i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.840462188827719874446867679209, −9.212879486606263588610389039542, −8.679389745117110926165653873899, −7.40192486737818341742223158679, −6.83237166736305553226023105055, −6.09797370327935468571856435792, −4.78199494654529479242810511242, −3.85728875618723341816388641216, −2.52381867671227407277573328748, −1.32990217166099898527693036394, 0.42027559159147366791037994264, 2.15700104508673027605128167644, 3.15630869334358871749293131531, 4.15497241411548340926350688589, 5.88960415092835783548164444614, 6.21493109140247131124802962384, 7.04444614829709831000873990967, 8.246795330553539931853325491137, 8.766752875597712008886375625666, 9.649022718894528597203314728578

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