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Properties

Label 2-1134-63.58-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.841 - 0.540i$
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 + (−0.5 + 2.59i)7-s + 8-s + (3 − 5.19i)11-s + (−2.5 + 4.33i)13-s + (−0.5 + 2.59i)14-s + 16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (3 − 5.19i)22-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (−2.5 + 4.33i)26-s + (−0.5 + 2.59i)28-s + (3 + 5.19i)29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (0.904 − 1.56i)11-s + (−0.693 + 1.20i)13-s + (−0.133 + 0.694i)14-s + 0.250·16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (0.639 − 1.10i)22-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (−0.490 + 0.849i)26-s + (−0.0944 + 0.490i)28-s + (0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.841 - 0.540i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.613231022\)
\(L(\frac12)\) \(\approx\) \(2.613231022\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.834200577861963653558935819176, −8.959865939426136703937909115793, −8.436941407549846736404055352110, −7.15235118997075856624656702725, −6.35293121075353093682762361401, −5.69751939932530624951526541798, −4.76574011115947488832893584227, −3.61524966323110508303152505132, −2.84334288646061969186461333611, −1.45685937729851190905882712233, 1.05265602894310832411085821794, 2.58924677193309664956985900145, 3.61238424405171559155945006030, 4.58314249904603549018329730152, 5.24357758870821828163612566850, 6.44416954844267361020173181362, 7.40621920555462411158286841358, 7.53110057991743244949354691771, 9.079530922070143779683080894501, 10.07410273012661055759578126232

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