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Properties

Label 2-1134-63.5-c1-0-28
Degree $2$
Conductor $1134$
Sign $-0.906 + 0.421i$
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  + i·2-s − 4-s + (−2.09 − 3.62i)5-s + (2.62 + 0.358i)7-s − i·8-s + (3.62 − 2.09i)10-s + (−2.59 − 1.5i)11-s + (−2.12 − 1.22i)13-s + (−0.358 + 2.62i)14-s + 16-s + (0.507 + 0.878i)17-s + (−0.878 − 0.507i)19-s + (2.09 + 3.62i)20-s + (1.5 − 2.59i)22-s + (−3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.935 − 1.61i)5-s + (0.990 + 0.135i)7-s − 0.353i·8-s + (1.14 − 0.661i)10-s + (−0.783 − 0.452i)11-s + (−0.588 − 0.339i)13-s + (−0.0958 + 0.700i)14-s + 0.250·16-s + (0.123 + 0.213i)17-s + (−0.201 − 0.116i)19-s + (0.467 + 0.809i)20-s + (0.319 − 0.553i)22-s + (−0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.906 + 0.421i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2660506786\)
\(L(\frac12)\) \(\approx\) \(0.2660506786\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.62 - 0.358i)T \)
good5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.507 - 0.878i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.07 - 0.621i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.61iT - 31T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 1.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.12 + 7.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.01T + 47T^{2} \)
53 \( 1 + (1.07 - 0.621i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 5.91iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + (1.58 + 2.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.25 + 1.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.057735977234583343623164254815, −8.433935992056534932232414066968, −7.962095170430365050677553378769, −7.30955123638238492602129073186, −5.81730152398410375380586127258, −5.02707678388376363398979926814, −4.64042537644860896572174980636, −3.46234660159862381242652839647, −1.57051582136444603426182023095, −0.11513540247021591870807620007, 2.06645675662271556107697841396, 2.84611754520478950508908171987, 3.98547222731230120802370385321, 4.67036738302943584160667344377, 5.94716973411456240358932183012, 7.16110005790491111263008568935, 7.66330760412475570306797010236, 8.355450291174159869264028357633, 9.681333101138082574682831751941, 10.40105060805185953924281934852

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