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Properties

Label 2-1134-63.25-c1-0-0
Degree $2$
Conductor $1134$
Sign $-0.995 + 0.0954i$
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 + (−1.5 + 2.59i)5-s + (−0.5 + 2.59i)7-s − 8-s + (1.5 − 2.59i)10-s + (−1.5 − 2.59i)11-s + (2 + 3.46i)13-s + (0.5 − 2.59i)14-s + 16-s + (2 + 3.46i)19-s + (−1.5 + 2.59i)20-s + (1.5 + 2.59i)22-s + (−2 − 3.46i)25-s + (−2 − 3.46i)26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.474 − 0.821i)10-s + (−0.452 − 0.783i)11-s + (0.554 + 0.960i)13-s + (0.133 − 0.694i)14-s + 0.250·16-s + (0.458 + 0.794i)19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s + (−0.400 − 0.692i)25-s + (−0.392 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.995 + 0.0954i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.995 + 0.0954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4620285812\)
\(L(\frac12)\) \(\approx\) \(0.4620285812\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40956159695169752535873812123, −9.193243255477770906332793676834, −8.753424907786203271168711792112, −7.74813541848390692302097948759, −7.07060616130216759071469563675, −6.19756851744592351201805173369, −5.41649508622841135193035146326, −3.72148327816219861880703108446, −3.06090563312400065949464149945, −1.86508777889752168606144419335, 0.26752694720351671425392445139, 1.34865547824049081645977225945, 3.01441604272877173242644037264, 4.19186594405240626134618734572, 4.95537758216251788004418144537, 6.10531205733708223827730759957, 7.29241025841412875729084723553, 7.80215690972555180573038961439, 8.477208284627670964360018491559, 9.434367834454531968786936231025

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