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Properties

Label 2-1134-63.58-c1-0-10
Degree $2$
Conductor $1134$
Sign $0.580 - 0.814i$
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 + (1 + 1.73i)5-s + (2 − 1.73i)7-s + 8-s + (1 + 1.73i)10-s + (−2.5 + 4.33i)11-s + (−3 + 5.19i)13-s + (2 − 1.73i)14-s + 16-s + (2 + 3.46i)17-s + (2 − 3.46i)19-s + (1 + 1.73i)20-s + (−2.5 + 4.33i)22-s + (2 + 3.46i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.447 + 0.774i)5-s + (0.755 − 0.654i)7-s + 0.353·8-s + (0.316 + 0.547i)10-s + (−0.753 + 1.30i)11-s + (−0.832 + 1.44i)13-s + (0.534 − 0.462i)14-s + 0.250·16-s + (0.485 + 0.840i)17-s + (0.458 − 0.794i)19-s + (0.223 + 0.387i)20-s + (−0.533 + 0.923i)22-s + (0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.580 - 0.814i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.772141299\)
\(L(\frac12)\) \(\approx\) \(2.772141299\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + (-3.5 - 6.06i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15965768758270726263802344904, −9.351765909098496548995232739890, −7.997528723035342657381666802011, −7.13447085635539441241188271660, −6.84759891320165157575489765626, −5.52911816155775780218357199961, −4.71931544280298242307784565299, −3.96789110106649266905388447973, −2.55823371600499328904540783514, −1.79510431774678117233425808359, 1.00973336661073530968017660656, 2.52422048446923244539279723219, 3.30981575671157054021274517671, 4.87778445035891662310190805752, 5.40545775814953363390883855875, 5.77583534029149056654467104280, 7.28294390040074188244438674279, 8.085853094283782705251715945283, 8.708561073442556179650331164240, 9.751629429527716385927317561787

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