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Properties

Label 2-1127-161.160-c1-0-60
Degree $2$
Conductor $1127$
Sign $-0.695 + 0.718i$
Analytic cond. $8.99914$
Root an. cond. $2.99985$
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  − 0.218·2-s − 2.36i·3-s − 1.95·4-s + 3.26·5-s + 0.515i·6-s + 0.862·8-s − 2.58·9-s − 0.711·10-s − 3.73i·11-s + 4.61i·12-s + 1.00i·13-s − 7.71i·15-s + 3.71·16-s − 4.44·17-s + 0.563·18-s + 5.90·19-s + ⋯
L(s)  = 1  − 0.154·2-s − 1.36i·3-s − 0.976·4-s + 1.45·5-s + 0.210i·6-s + 0.304·8-s − 0.861·9-s − 0.225·10-s − 1.12i·11-s + 1.33i·12-s + 0.277i·13-s − 1.99i·15-s + 0.929·16-s − 1.07·17-s + 0.132·18-s + 1.35·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(8.99914\)
Root analytic conductor: \(2.99985\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (1126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.411202956\)
\(L(\frac12)\) \(\approx\) \(1.411202956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 + (2.88 + 3.83i)T \)
good2 \( 1 + 0.218T + 2T^{2} \)
3 \( 1 + 2.36iT - 3T^{2} \)
5 \( 1 - 3.26T + 5T^{2} \)
11 \( 1 + 3.73iT - 11T^{2} \)
13 \( 1 - 1.00iT - 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + 4.67iT - 31T^{2} \)
37 \( 1 + 0.741iT - 37T^{2} \)
41 \( 1 + 5.56iT - 41T^{2} \)
43 \( 1 - 2.01iT - 43T^{2} \)
47 \( 1 + 2.83iT - 47T^{2} \)
53 \( 1 - 5.48iT - 53T^{2} \)
59 \( 1 - 13.8iT - 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 4.07iT - 73T^{2} \)
79 \( 1 - 6.11iT - 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.369537516005196719061451668052, −8.724436350484509939250243117789, −7.962250979603743682193281432136, −6.96287246187308643522522625275, −6.07125108503936924611603396482, −5.59867322799011906999053740622, −4.37631921257719827049555206975, −2.85942207045335663593818702469, −1.77920805407606082116732370685, −0.70090364211857850903011793116, 1.63483935956016531701477892880, 3.09406132287598911044813605890, 4.23377774637318763252467220395, 4.99980785655527359984246398247, 5.48099987010393805883527512559, 6.65613807258538307915780673905, 7.87609252452464360064079430647, 9.027857805560035941833784402241, 9.420696646396144278891289868203, 10.02725539504693039160543753385

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