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Properties

Label 2-1134-7.4-c1-0-21
Degree $2$
Conductor $1134$
Sign $-0.574 + 0.818i$
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 + (0.230 + 0.398i)5-s + (−2.32 + 1.26i)7-s + 0.999·8-s + (0.230 − 0.398i)10-s + (1.82 − 3.15i)11-s − 1.46·13-s + (2.25 + 1.38i)14-s + (−0.5 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (−2.02 − 3.51i)19-s − 0.460·20-s − 3.64·22-s + (−0.566 − 0.981i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.102 + 0.178i)5-s + (−0.878 + 0.478i)7-s + 0.353·8-s + (0.0728 − 0.126i)10-s + (0.549 − 0.952i)11-s − 0.405·13-s + (0.603 + 0.368i)14-s + (−0.125 − 0.216i)16-s + (−0.452 + 0.784i)17-s + (−0.465 − 0.805i)19-s − 0.102·20-s − 0.777·22-s + (−0.118 − 0.204i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7990962399\)
\(L(\frac12)\) \(\approx\) \(0.7990962399\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 1.26i)T \)
good5 \( 1 + (-0.230 - 0.398i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.82 + 3.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.566 + 0.981i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.97T + 29T^{2} \)
31 \( 1 + (-0.257 + 0.445i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.945T + 41T^{2} \)
43 \( 1 + 9.32T + 43T^{2} \)
47 \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.64T + 83T^{2} \)
89 \( 1 + (1.36 + 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−9.621004341105030715166194494771, −8.644240158644945047176214270723, −8.380054230551556400592415574403, −6.78439226911131595715010736153, −6.43196961011825598775704146582, −5.19519306690671494788902375234, −4.00584817743247724348132592096, −3.07625436060763517994838722589, −2.15340400706614926263789008071, −0.41508531589434567343333966809, 1.33172345444547014894403688882, 2.88533625539755533063736273890, 4.19448576483144773271991219079, 4.96299467513367235472222762136, 6.14085776457858723784441079590, 6.88601878420812743251630708524, 7.40378649262646684649841589186, 8.555801165934233218999919145966, 9.248191055705522846102719016229, 10.05604508814928196186744945862

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