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Properties

Label 2-1134-7.4-c1-0-22
Degree $2$
Conductor $1134$
Sign $-0.0996 + 0.995i$
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 + (0.794 + 1.37i)5-s + (1.40 − 2.24i)7-s + 0.999·8-s + (0.794 − 1.37i)10-s + (−0.794 + 1.37i)11-s − 4.81·13-s + (−2.64 − 0.0963i)14-s + (−0.5 − 0.866i)16-s + (2.69 − 4.67i)17-s + (−3.54 − 6.14i)19-s − 1.58·20-s + 1.58·22-s + (0.150 + 0.260i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.355 + 0.615i)5-s + (0.531 − 0.847i)7-s + 0.353·8-s + (0.251 − 0.434i)10-s + (−0.239 + 0.414i)11-s − 1.33·13-s + (−0.706 − 0.0257i)14-s + (−0.125 − 0.216i)16-s + (0.654 − 1.13i)17-s + (−0.814 − 1.41i)19-s − 0.355·20-s + 0.338·22-s + (0.0313 + 0.0542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0996 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0996 + 0.995i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.0996 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.241471525\)
\(L(\frac12)\) \(\approx\) \(1.241471525\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 2.24i)T \)
good5 \( 1 + (-0.794 - 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.794 - 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.150 - 0.260i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.87T + 41T^{2} \)
43 \( 1 - 1.66T + 43T^{2} \)
47 \( 1 + (-1.33 - 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.23 + 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.36T + 83T^{2} \)
89 \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.42T + 97T^{2} \)
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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

−9.825136783074864612478742179300, −8.990573689037470029667955284079, −7.83316609676530799339298312192, −7.27987648469729519395109807142, −6.49065059402026926574467265170, −4.91762161926709045036768818197, −4.48882644984893474892041174544, −2.93836736713722294963511673337, −2.29968485884261114315892631214, −0.65279016507163005034804944902, 1.36749441425256368159136127830, 2.55406322785989776867499544300, 4.16115206835326002881985035987, 5.21392178162771262672040489475, 5.70052166827013595159114961207, 6.63395786102189213780338882454, 7.85850271715080812272828901619, 8.326473242853281587314569099089, 9.021654091378583787947067532061, 9.980888718673977003921097039877

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