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Properties

Label 2-1134-21.5-c1-0-15
Degree $2$
Conductor $1134$
Sign $0.585 - 0.810i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.860 − 1.49i)5-s + (−2.25 + 1.37i)7-s + 0.999i·8-s + (1.49 − 0.860i)10-s + (1.47 − 0.854i)11-s + 1.28i·13-s + (−2.64 + 0.0645i)14-s + (−0.5 + 0.866i)16-s + (2.60 + 4.50i)17-s + (2.39 + 1.38i)19-s + 1.72·20-s + 1.70·22-s + (4.71 + 2.72i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.384 − 0.666i)5-s + (−0.853 + 0.520i)7-s + 0.353i·8-s + (0.471 − 0.272i)10-s + (0.445 − 0.257i)11-s + 0.355i·13-s + (−0.706 + 0.0172i)14-s + (−0.125 + 0.216i)16-s + (0.630 + 1.09i)17-s + (0.549 + 0.317i)19-s + 0.384·20-s + 0.364·22-s + (0.983 + 0.567i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.374778245\)
\(L(\frac12)\) \(\approx\) \(2.374778245\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.25 - 1.37i)T \)
good5 \( 1 + (-0.860 + 1.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.47 + 0.854i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.28iT - 13T^{2} \)
17 \( 1 + (-2.60 - 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.39 - 1.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.71 - 2.72i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.44iT - 29T^{2} \)
31 \( 1 + (-5.82 + 3.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 + 6.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.96T + 41T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 + (2.61 - 4.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.46 - 5.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.28 + 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.31 + 7.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.07iT - 71T^{2} \)
73 \( 1 + (-8.18 + 4.72i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.73 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.07T + 83T^{2} \)
89 \( 1 + (-4.16 + 7.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−9.634951287743954390100458305086, −9.218230972432012615082591901450, −8.294098356783611717622871766802, −7.36325800297759821285164475138, −6.27711457181521389661389652671, −5.82500836249266292211370480194, −4.89279363624209462024345757079, −3.79760706071796188276025290989, −2.90208855367359693225273474833, −1.41007057972973329524827859369, 0.974548884073076628261355121287, 2.73387783676198825115296693345, 3.18832504526971270043122511243, 4.44018048158738142980988877943, 5.33255246332446781078554588188, 6.56005372862021518520278501309, 6.77330257535449860859287305971, 7.88126886135293125973420548350, 9.203557567830748726567910415697, 9.924273009306470858385501218069

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