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Properties

Label 2-1134-63.4-c1-0-27
Degree $2$
Conductor $1134$
Sign $-0.678 + 0.734i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 + 1.73i)7-s + 0.999·8-s + (2 − 3.46i)13-s + (−0.499 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s + (−1 − 1.73i)19-s − 3·23-s − 5·25-s + (1.99 + 3.46i)26-s + (2.49 + 0.866i)28-s + (3 + 5.19i)29-s + (−2.5 − 4.33i)31-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 + 0.654i)7-s + 0.353·8-s + (0.554 − 0.960i)13-s + (−0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s − 0.625·23-s − 25-s + (0.392 + 0.679i)26-s + (0.472 + 0.163i)28-s + (0.557 + 0.964i)29-s + (−0.449 − 0.777i)31-s + (−0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.321625853821069835027506088770, −8.671512861062864257852877442902, −7.984718837291666890931885437457, −6.97326153982318964600085723884, −6.04194366736133261960039293027, −5.66320204799691869378309545078, −4.30772461514516820171622143200, −3.25213847503822875344463596991, −1.88626956593863709125787110557, 0, 1.60666715479128206862102868690, 2.87457999811761605231113501978, 3.89731672134636019352475857981, 4.64545205436501340054512287153, 6.08853665698931840005713745024, 6.85029001526945372571471571174, 7.69427999619948725340724044947, 8.704513245761455668575784358552, 9.415507279830890488392113325475

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