Properties

Label 2-1134-63.25-c1-0-5
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 + (−0.822 + 1.42i)5-s + (−1.32 + 2.29i)7-s + 8-s + (−0.822 + 1.42i)10-s + (0.822 + 1.42i)11-s + (−0.322 − 0.559i)13-s + (−1.32 + 2.29i)14-s + 16-s + (0.822 − 1.42i)17-s + (−1 − 1.73i)19-s + (−0.822 + 1.42i)20-s + (0.822 + 1.42i)22-s + (−4.64 + 8.04i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.368 + 0.637i)5-s + (−0.499 + 0.866i)7-s + 0.353·8-s + (−0.260 + 0.450i)10-s + (0.248 + 0.429i)11-s + (−0.0895 − 0.155i)13-s + (−0.353 + 0.612i)14-s + 0.250·16-s + (0.199 − 0.345i)17-s + (−0.229 − 0.397i)19-s + (−0.184 + 0.318i)20-s + (0.175 + 0.303i)22-s + (−0.968 + 1.67i)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

\(L(1)\) \(\approx\) \(1.830970323\)
\(L(\frac12)\) \(\approx\) \(1.830970323\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.822 - 1.42i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.822 - 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.322 + 0.559i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.822 + 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.64 - 8.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.82 - 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.645T + 31T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.46 - 4.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 8.29T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-5.29 + 9.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + (-1.35 + 2.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.46 - 9.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.79 + 6.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.971231222042208491735339642190, −9.432043802753550003327154789148, −8.314933301663733298765137964615, −7.32604514184740708157278796666, −6.72479872903577734837431487349, −5.74137361978059417326908259122, −5.01705651591322004394733772630, −3.71470549362742452654836125452, −3.06902715431995479093364851583, −1.89499811494096512355266646600, 0.61280032077115953288509947455, 2.23339136183512520727868701702, 3.69504557770311345886382816053, 4.14386015456911545012253653786, 5.16358319337739047583062230940, 6.27278171829400630598008707891, 6.82354877865124450663086070398, 8.031015666023762156749177255981, 8.489182041255453950120975593828, 9.821472477169078537900788753684

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