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Properties

Label 2-1134-63.25-c1-0-30
Degree $2$
Conductor $1134$
Sign $0.281 + 0.959i$
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 + (1.82 − 3.15i)5-s + (1.32 − 2.29i)7-s + 8-s + (1.82 − 3.15i)10-s + (−1.82 − 3.15i)11-s + (2.32 + 4.02i)13-s + (1.32 − 2.29i)14-s + 16-s + (−1.82 + 3.15i)17-s + (−1 − 1.73i)19-s + (1.82 − 3.15i)20-s + (−1.82 − 3.15i)22-s + (0.645 − 1.11i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.815 − 1.41i)5-s + (0.499 − 0.866i)7-s + 0.353·8-s + (0.576 − 0.998i)10-s + (−0.549 − 0.951i)11-s + (0.644 + 1.11i)13-s + (0.353 − 0.612i)14-s + 0.250·16-s + (−0.442 + 0.765i)17-s + (−0.229 − 0.397i)19-s + (0.407 − 0.705i)20-s + (−0.388 − 0.673i)22-s + (0.134 − 0.233i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.281 + 0.959i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.961204372\)
\(L(\frac12)\) \(\approx\) \(2.961204372\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.82 + 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.32 - 4.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.82 - 3.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.645 + 1.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.17 - 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + (-5.96 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.46 + 9.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.93T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.35T + 59T^{2} \)
61 \( 1 - 7.35T + 61T^{2} \)
67 \( 1 + 2.29T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + (5.29 - 9.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-6.64 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.46 + 4.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.79 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.617187399653750554504430143267, −8.670478135123780277606429486420, −8.234516653962075939173444573956, −6.93839933724472584700814318179, −6.09961200850969435708318059726, −5.24400367154001425098909853770, −4.52545422821642011329074667281, −3.69892932677045210678297604870, −2.07312648661058247143207900983, −1.08685075665687360093488358958, 2.04874632403757910557576302031, 2.61760043727950654055780612415, 3.66990128970961452086593701665, 5.08872631053590079798610718286, 5.67468550254103755322802732186, 6.49150907659239550835565813387, 7.34150179883781074474635831512, 8.127726220401609456371681269668, 9.395847763302709396194252148266, 10.10867640786596732088284455408

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