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Properties

Label 2-1134-63.25-c1-0-31
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 $1$

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: $\chi_{1134} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.439222300457382733408869521821, −8.243743845843407785255605399231, −7.944767899578993650195622107364, −7.04308292674896108217920689708, −5.94909782720694560338006323901, −5.28008244412773801919236121150, −3.70567316124604429688224454637, −3.02496571441524899250316319663, −1.39983188152735632284314568772, 0, 2.28736440293479009498956791580, 2.47117374906047835905413622225, 4.45347836580512473922504394836, 5.08151176843943850475549135665, 6.38412075671998449833746019009, 7.08586115371852596081954552463, 7.79971639130881510761298207324, 8.908822312768315671147402873206, 9.436478990249032148798259479475

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