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Properties

Label 2-1134-21.17-c1-0-29
Degree $2$
Conductor $1134$
Sign $-0.939 + 0.341i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.08 − 3.61i)5-s + (1.94 − 1.79i)7-s − 0.999i·8-s + (−3.61 − 2.08i)10-s + (−0.0814 − 0.0470i)11-s − 3.79i·13-s + (0.786 − 2.52i)14-s + (−0.5 − 0.866i)16-s + (−3.82 + 6.62i)17-s + (6.77 − 3.91i)19-s − 4.17·20-s − 0.0940·22-s + (−3.31 + 1.91i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.932 − 1.61i)5-s + (0.734 − 0.678i)7-s − 0.353i·8-s + (−1.14 − 0.659i)10-s + (−0.0245 − 0.0141i)11-s − 1.05i·13-s + (0.210 − 0.675i)14-s + (−0.125 − 0.216i)16-s + (−0.928 + 1.60i)17-s + (1.55 − 0.897i)19-s − 0.932·20-s − 0.0200·22-s + (−0.691 + 0.399i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.939 + 0.341i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.939 + 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.770630983\)
\(L(\frac12)\) \(\approx\) \(1.770630983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.94 + 1.79i)T \)
good5 \( 1 + (2.08 + 3.61i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.0814 + 0.0470i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.79iT - 13T^{2} \)
17 \( 1 + (3.82 - 6.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.77 + 3.91i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.31 - 1.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.03iT - 29T^{2} \)
31 \( 1 + (1.45 + 0.840i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.18 + 3.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.07T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 + (2.54 + 4.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.05 - 3.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.21 + 9.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.69 + 1.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.76 - 6.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.61iT - 71T^{2} \)
73 \( 1 + (0.129 + 0.0750i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.48 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.58T + 83T^{2} \)
89 \( 1 + (1.37 + 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.06iT - 97T^{2} \)
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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

−9.428152974441929753395842300863, −8.473976125094007429555213619258, −7.950743743197033799278028395273, −7.08987434321144960931733116827, −5.60145036018356978521829696654, −5.03479626642602398934293372861, −4.18238731472704787402975353818, −3.52268481603683622277641199917, −1.71951048314237182701815080214, −0.63933299752059972166933218157, 2.22179823790831670746866091927, 3.08438173384437254749307635025, 4.07328910240226976936924634909, 4.97638077649385489460507660466, 6.08710799357097018646040231823, 6.93916414402454978548125335353, 7.47237146741230075815074855550, 8.249676296258074692032487565154, 9.324246771321844227332362142733, 10.33140922519681049020652543780

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