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Properties

Label 2-1134-9.2-c2-0-10
Degree $2$
Conductor $1134$
Sign $-0.766 - 0.642i$
Analytic cond. $30.8992$
Root an. cond. $5.55871$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−7.70 + 4.44i)5-s + (−1.32 + 2.29i)7-s − 2.82i·8-s + 12.5·10-s + (15.4 + 8.89i)11-s + (−1.29 − 2.23i)13-s + (3.24 − 1.87i)14-s + (−2.00 + 3.46i)16-s + 25.8i·17-s + 20·19-s + (−15.4 − 8.89i)20-s + (−12.5 − 21.7i)22-s + (−15.4 + 8.89i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.54 + 0.889i)5-s + (−0.188 + 0.327i)7-s − 0.353i·8-s + 1.25·10-s + (1.40 + 0.808i)11-s + (−0.0993 − 0.172i)13-s + (0.231 − 0.133i)14-s + (−0.125 + 0.216i)16-s + 1.52i·17-s + 1.05·19-s + (−0.770 − 0.444i)20-s + (−0.571 − 0.990i)22-s + (−0.670 + 0.386i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(30.8992\)
Root analytic conductor: \(5.55871\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7621197036\)
\(L(\frac12)\) \(\approx\) \(0.7621197036\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (1.32 - 2.29i)T \)
good5 \( 1 + (7.70 - 4.44i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-15.4 - 8.89i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.29 + 2.23i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 25.8iT - 289T^{2} \)
19 \( 1 - 20T + 361T^{2} \)
23 \( 1 + (15.4 - 8.89i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.3 - 5.95i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.58 - 14.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 + (-13.6 + 7.86i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (14.6 + 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 85.5iT - 2.80e3T^{2} \)
59 \( 1 + (1.42 - 0.824i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (50.1 - 86.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (18.3 + 31.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 17.7iT - 5.04e3T^{2} \)
73 \( 1 - 28.9T + 5.32e3T^{2} \)
79 \( 1 + (59.1 - 102. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (104. + 60.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 139. iT - 7.92e3T^{2} \)
97 \( 1 + (22.2 - 38.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.00089374233622022968832964400, −9.127686924801508290954048797050, −8.264538753648643466486263774811, −7.52800857658912140528530945654, −6.89964283701394418658232891328, −6.00156403442023568147255024343, −4.27969712116180669253123690886, −3.75097800675615115826273140882, −2.76233669969794867741667310890, −1.31019503050528938947396341197, 0.37306348615541258959608951117, 1.10038652688314184322251737233, 3.10775372763943176427501568292, 4.11402159359041017093236518420, 4.86625776108268937641979413436, 6.10520743887222391287617720756, 7.04813095079627867100979636023, 7.76297834060130230150504921532, 8.407497234981191906346906472294, 9.262860484720248957065226221699

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