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Properties

Label 2-1134-63.47-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.477 + 0.878i$
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 − 3.50·5-s + (−2.27 + 1.35i)7-s + 0.999i·8-s + (3.03 − 1.75i)10-s + 5.45i·11-s + (2.18 − 1.25i)13-s + (1.29 − 2.30i)14-s + (−0.5 − 0.866i)16-s + (−0.852 − 1.47i)17-s + (−4.19 − 2.42i)19-s + (−1.75 + 3.03i)20-s + (−2.72 − 4.72i)22-s + 5.35i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.56·5-s + (−0.859 + 0.511i)7-s + 0.353i·8-s + (0.960 − 0.554i)10-s + 1.64i·11-s + (0.604 − 0.349i)13-s + (0.345 − 0.617i)14-s + (−0.125 − 0.216i)16-s + (−0.206 − 0.358i)17-s + (−0.963 − 0.556i)19-s + (−0.392 + 0.679i)20-s + (−0.581 − 1.00i)22-s + 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3249713602\)
\(L(\frac12)\) \(\approx\) \(0.3249713602\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 - 1.35i)T \)
good5 \( 1 + 3.50T + 5T^{2} \)
11 \( 1 - 5.45iT - 11T^{2} \)
13 \( 1 + (-2.18 + 1.25i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.852 + 1.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.19 + 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.35iT - 23T^{2} \)
29 \( 1 + (6.16 + 3.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.25 - 3.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.14 + 7.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.37 + 4.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.22 - 5.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.31 + 7.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.0 + 5.81i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.94 + 6.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.26 + 4.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.895 + 1.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.791iT - 71T^{2} \)
73 \( 1 + (11.5 - 6.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.40 + 4.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.232 + 0.403i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.09 + 1.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 + 2.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.597170207186921077753465365002, −8.773557182350189281411292311526, −8.028784023193918146154779118275, −7.19422788608462363684069927083, −6.71732690262810092819820471672, −5.48338552487562959956801082183, −4.38655988451740628931578744803, −3.50645288376414795026914979892, −2.17352375905286295814707365544, −0.23470594531992026813081689430, 0.906456401972270886309493750613, 2.88604726416725383314461345669, 3.72064339765925297348457170617, 4.28228874269813725512434932114, 6.05731958375198548920664468655, 6.70719679764625780151510665747, 7.72101449140272369705751778169, 8.472067740118638332375557925082, 8.824769086348946153173197117217, 10.14848785810739830168566685104

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