Properties

Label 2-1134-21.20-c1-0-28
Degree $2$
Conductor $1134$
Sign $-0.407 + 0.913i$
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  i·2-s − 4-s + 2.34·5-s + (1.07 − 2.41i)7-s + i·8-s − 2.34i·10-s − 5.67i·11-s + 1.71i·13-s + (−2.41 − 1.07i)14-s + 16-s + 1.76·17-s − 1.13i·19-s − 2.34·20-s − 5.67·22-s + 3.67i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.05·5-s + (0.407 − 0.913i)7-s + 0.353i·8-s − 0.742i·10-s − 1.71i·11-s + 0.477i·13-s + (−0.645 − 0.288i)14-s + 0.250·16-s + 0.429·17-s − 0.261i·19-s − 0.525·20-s − 1.21·22-s + 0.766i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.407 + 0.913i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.856948066\)
\(L(\frac12)\) \(\approx\) \(1.856948066\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-1.07 + 2.41i)T \)
good5 \( 1 - 2.34T + 5T^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
13 \( 1 - 1.71iT - 13T^{2} \)
17 \( 1 - 1.76T + 17T^{2} \)
19 \( 1 + 1.13iT - 19T^{2} \)
23 \( 1 - 3.67iT - 23T^{2} \)
29 \( 1 - 4.15iT - 29T^{2} \)
31 \( 1 + 8.37iT - 31T^{2} \)
37 \( 1 + 9.19T + 37T^{2} \)
41 \( 1 - 7.99T + 41T^{2} \)
43 \( 1 + 3.52T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 2.22T + 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + 4.52iT - 71T^{2} \)
73 \( 1 - 5.34iT - 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 - 4.59iT - 97T^{2} \)
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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

−9.660578052876508185548332975780, −8.935759705641816727522455785936, −8.089894738198778589911148587298, −7.07805131687448432074096357387, −5.94257854542717655622363174415, −5.34607586131042199687986658344, −4.09887237019612837037939333769, −3.24176733616498206097477587719, −1.97656993578732109411043202170, −0.857797693006781451123261729050, 1.67318351653426810046563007841, 2.64437977367793050065619021444, 4.26932248717629533912599724840, 5.22641679562659197562609627718, 5.74251514993785165273328401174, 6.72666431570743034822573767704, 7.51531822309151793776597408253, 8.484368275297951523788194646429, 9.186481128162343937222456108824, 9.997393913591396561304360265663

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