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Properties

Label 2-1127-161.3-c0-0-0
Degree $2$
Conductor $1127$
Sign $-0.462 + 0.886i$
Analytic cond. $0.562446$
Root an. cond. $0.749964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.50 − 1.18i)2-s + (0.632 − 2.60i)4-s + (−1.34 − 2.93i)8-s + (−0.327 + 0.945i)9-s + (−0.653 − 0.513i)11-s + (−3.12 − 1.60i)16-s + (0.627 + 1.81i)18-s − 1.59·22-s + (0.981 + 0.189i)23-s + (0.928 + 0.371i)25-s + (0.273 + 0.0801i)29-s + (−3.44 + 0.664i)32-s + (2.25 + 1.45i)36-s + (−0.550 + 1.58i)37-s + (−0.544 + 1.19i)43-s + (−1.75 + 1.37i)44-s + ⋯
L(s)  = 1  + (1.50 − 1.18i)2-s + (0.632 − 2.60i)4-s + (−1.34 − 2.93i)8-s + (−0.327 + 0.945i)9-s + (−0.653 − 0.513i)11-s + (−3.12 − 1.60i)16-s + (0.627 + 1.81i)18-s − 1.59·22-s + (0.981 + 0.189i)23-s + (0.928 + 0.371i)25-s + (0.273 + 0.0801i)29-s + (−3.44 + 0.664i)32-s + (2.25 + 1.45i)36-s + (−0.550 + 1.58i)37-s + (−0.544 + 1.19i)43-s + (−1.75 + 1.37i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ -0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.166734829\)
\(L(\frac12)\) \(\approx\) \(2.166734829\)
\(L(1)\) 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 \( 1 + (-0.981 - 0.189i)T \)
good2 \( 1 + (-1.50 + 1.18i)T + (0.235 - 0.971i)T^{2} \)
3 \( 1 + (0.327 - 0.945i)T^{2} \)
5 \( 1 + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.0475 + 0.998i)T^{2} \)
19 \( 1 + (-0.0475 - 0.998i)T^{2} \)
29 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (-0.981 - 0.189i)T^{2} \)
37 \( 1 + (0.550 - 1.58i)T + (-0.786 - 0.618i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.0135 + 0.284i)T + (-0.995 + 0.0950i)T^{2} \)
59 \( 1 + (-0.580 + 0.814i)T^{2} \)
61 \( 1 + (0.327 + 0.945i)T^{2} \)
67 \( 1 + (1.78 + 0.713i)T + (0.723 + 0.690i)T^{2} \)
71 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.888 + 0.458i)T^{2} \)
79 \( 1 + (0.0135 - 0.284i)T + (-0.995 - 0.0950i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (-0.981 + 0.189i)T^{2} \)
97 \( 1 + (0.142 + 0.989i)T^{2} \)
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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

−10.28594287660078257479863710406, −9.234160759326829274576350425656, −8.155628571223475129004486822576, −6.92885273848245166747990047922, −5.97734376810402675638248821633, −5.08120266492599458417377440190, −4.68260706031711349690108405013, −3.26880390614205131684428745042, −2.75254036681795501308162052269, −1.47669665814270921933352119751, 2.57437384039313020388566752442, 3.45570445423395629131178945840, 4.41703500435576621049991599536, 5.24008749437731047217468126920, 5.98366452303404522029924652680, 6.89773818281505889556336627348, 7.35674945918912766232064473122, 8.457660111961138907653648283800, 9.060629186595598492397508690873, 10.43448655596122400110413422963

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