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Properties

Label 2-1134-63.58-c1-0-24
Degree $2$
Conductor $1134$
Sign $0.585 + 0.810i$
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  + 2-s + 4-s + (−2.62 − 0.358i)7-s + 8-s + (2.12 − 3.67i)11-s + (1.12 − 1.94i)13-s + (−2.62 − 0.358i)14-s + 16-s + (1.12 − 1.94i)19-s + (2.12 − 3.67i)22-s + (0.621 + 1.07i)23-s + (2.5 − 4.33i)25-s + (1.12 − 1.94i)26-s + (−2.62 − 0.358i)28-s + (−2.12 − 3.67i)29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.990 − 0.135i)7-s + 0.353·8-s + (0.639 − 1.10i)11-s + (0.310 − 0.538i)13-s + (−0.700 − 0.0958i)14-s + 0.250·16-s + (0.257 − 0.445i)19-s + (0.452 − 0.783i)22-s + (0.129 + 0.224i)23-s + (0.5 − 0.866i)25-s + (0.219 − 0.380i)26-s + (−0.495 − 0.0677i)28-s + (−0.393 − 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.291888556\)
\(L(\frac12)\) \(\approx\) \(2.291888556\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (2.62 + 0.358i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.12 + 1.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.621 - 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.12 + 3.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.24T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.74 - 9.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.24 + 9.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.75T + 47T^{2} \)
53 \( 1 + (-2.12 - 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2.24T + 61T^{2} \)
67 \( 1 - 0.242T + 67T^{2} \)
71 \( 1 - 1.24T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + (8.12 + 14.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.74 - 9.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.24 + 3.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.820936283611722993341788435210, −8.839518622338183084473200526243, −8.049812709742009168599123508665, −6.88316519736778680718008100894, −6.28027290242232149577510658231, −5.55261363985728242311238557284, −4.35260855143770727217748518036, −3.43428084609333900724449221580, −2.69416100570659515776841478930, −0.848425312752751112604036672503, 1.55195781323104486028585875814, 2.86599233286990464963645757352, 3.80201395132030358330795139615, 4.68212732373341461212580019369, 5.69912368470237626215955805653, 6.71981014442410127222451658635, 7.00470424089250512180215090378, 8.286386701447857567209420377790, 9.307715199850748306253004176438, 9.883625602521738445347318549605

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