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Properties

Label 2-1134-7.2-c1-0-10
Degree $2$
Conductor $1134$
Sign $0.947 + 0.318i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.62 + 0.358i)7-s − 0.999·8-s + (2.12 + 3.67i)11-s + 6.24·13-s + (−1 + 2.44i)14-s + (−0.5 + 0.866i)16-s + (−3.12 + 5.40i)19-s + 4.24·22-s + (3.62 − 6.27i)23-s + (2.5 + 4.33i)25-s + (3.12 − 5.40i)26-s + (1.62 + 2.09i)28-s + 4.24·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (0.639 + 1.10i)11-s + 1.73·13-s + (−0.267 + 0.654i)14-s + (−0.125 + 0.216i)16-s + (−0.716 + 1.24i)19-s + 0.904·22-s + (0.755 − 1.30i)23-s + (0.5 + 0.866i)25-s + (0.612 − 1.06i)26-s + (0.306 + 0.395i)28-s + 0.787·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.835789525\)
\(L(\frac12)\) \(\approx\) \(1.835789525\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.358i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 - 5.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.62 + 6.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + (0.378 + 0.655i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.48T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + (-6.62 + 11.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.12 - 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.12 - 5.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.12 - 7.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.62 - 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.75T + 83T^{2} \)
89 \( 1 + (2.74 - 4.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.948694456731878107414989350971, −9.003087688266583542188114542851, −8.464240266369483732800051568239, −7.00019124655823717169164130997, −6.38705832407178701061822101900, −5.54008639792809073220520007301, −4.22366056988057120209760667907, −3.66791499865238834040907501815, −2.47712399365658871957802800430, −1.18482598715992105301683389016, 0.900706763106826821462493922292, 2.97919210318452889482327971902, 3.64294174006027829471440990867, 4.66644832906365654022997466236, 6.00147610301634692981211214369, 6.31071868127956108033554638464, 7.12956742297784514000860845186, 8.375585436914195344979363972807, 8.847468258435587034062028263559, 9.623069187964375689091441357827

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