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Properties

Label 2-1134-9.7-c1-0-4
Degree $2$
Conductor $1134$
Sign $0.173 - 0.984i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s + 3·10-s + (−1.5 + 2.59i)11-s + (2 + 3.46i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 6·17-s − 7·19-s + (1.50 − 2.59i)20-s + (1.5 + 2.59i)22-s + (1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.188 + 0.327i)7-s − 0.353·8-s + 0.948·10-s + (−0.452 + 0.783i)11-s + (0.554 + 0.960i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s − 1.45·17-s − 1.60·19-s + (0.335 − 0.580i)20-s + (0.319 + 0.553i)22-s + (0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.457470188\)
\(L(\frac12)\) \(\approx\) \(1.457470188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.21573703022119596023887242263, −9.294385969806531741035839376069, −8.682706816228100642796548089709, −7.23127755651231028803050604544, −6.54074369264001809800659256991, −5.89162690390824851179670978584, −4.64501113420199188984415023813, −3.79421652203593157403378953867, −2.42810963841338972513826963496, −2.05164060815842562480150541531, 0.52708159331006191865992390190, 2.21237225023772979770243576229, 3.61397700878471967169425439184, 4.62988275667730495171662040456, 5.39013433183998221583354016951, 6.16837736391229946838681691161, 6.94665877024398364258631591236, 8.299611877144388942130204100971, 8.580036781848298818569007847199, 9.339331760542930896811336916456

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