Properties

Label 2-1134-21.5-c1-0-14
Degree $2$
Conductor $1134$
Sign $0.248 - 0.968i$
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 + (0.714 − 1.23i)5-s + (2.10 + 1.59i)7-s + 0.999i·8-s + (1.23 − 0.714i)10-s + (−2.96 + 1.70i)11-s + 6.33i·13-s + (1.02 + 2.43i)14-s + (−0.5 + 0.866i)16-s + (−1.14 − 1.97i)17-s + (−1.87 − 1.08i)19-s + 1.42·20-s − 3.41·22-s + (6.97 + 4.02i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.319 − 0.553i)5-s + (0.797 + 0.603i)7-s + 0.353i·8-s + (0.391 − 0.226i)10-s + (−0.892 + 0.515i)11-s + 1.75i·13-s + (0.275 + 0.651i)14-s + (−0.125 + 0.216i)16-s + (−0.276 − 0.479i)17-s + (−0.430 − 0.248i)19-s + 0.319·20-s − 0.729·22-s + (1.45 + 0.839i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.248 - 0.968i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.480218835\)
\(L(\frac12)\) \(\approx\) \(2.480218835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 1.59i)T \)
good5 \( 1 + (-0.714 + 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.33iT - 13T^{2} \)
17 \( 1 + (1.14 + 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.345iT - 29T^{2} \)
31 \( 1 + (-3.76 + 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.404T + 41T^{2} \)
43 \( 1 + 5.81T + 43T^{2} \)
47 \( 1 + (-2.75 + 4.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.56 + 4.94i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.51 + 9.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.94 - 5.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.12 + 3.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 + (0.201 - 0.116i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.28 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.62T + 83T^{2} \)
89 \( 1 + (2.02 - 3.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−9.810142435756915890891133685814, −9.013259949828629302321263832006, −8.427118775990803060691043896244, −7.31685793032797969211842374969, −6.70802131943269556589176603919, −5.42973556995801875371260059434, −4.98270702865456352626905525201, −4.17483548204290258776697585017, −2.64369074809063038105543940534, −1.70864115957055169077426865473, 0.929051356261460369757154142425, 2.51661213806334950696614473367, 3.20917880285963060472759067117, 4.48896711936634848722722447613, 5.27188318930273238865717780272, 6.11432077943707475434492671431, 7.07810332047636938851315460160, 8.019551832395940891961274185593, 8.654968114077412397581611901769, 10.20220775192408686901070607383

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