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Properties

Label 2-1127-161.130-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.388 + 0.921i$
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  + (1.78 − 0.713i)2-s + (1.94 − 1.85i)4-s + (1.34 − 2.93i)8-s + (−0.981 − 0.189i)9-s + (−0.676 + 1.68i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s + 3.49i·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (−1.14 − 3.32i)32-s + (−2.25 + 1.45i)36-s + (−0.204 + 1.06i)37-s + (1.37 − 0.627i)43-s + (1.81 + 4.53i)44-s + ⋯
L(s)  = 1  + (1.78 − 0.713i)2-s + (1.94 − 1.85i)4-s + (1.34 − 2.93i)8-s + (−0.981 − 0.189i)9-s + (−0.676 + 1.68i)11-s + (0.167 − 3.50i)16-s + (−1.88 + 0.363i)18-s + 3.49i·22-s + (−0.327 + 0.945i)23-s + (−0.786 − 0.618i)25-s + (0.273 − 0.0801i)29-s + (−1.14 − 3.32i)32-s + (−2.25 + 1.45i)36-s + (−0.204 + 1.06i)37-s + (1.37 − 0.627i)43-s + (1.81 + 4.53i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.607147787\)
\(L(\frac12)\) \(\approx\) \(2.607147787\)
\(L(1)\) 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 \( 1 + (0.327 - 0.945i)T \)
good2 \( 1 + (-1.78 + 0.713i)T + (0.723 - 0.690i)T^{2} \)
3 \( 1 + (0.981 + 0.189i)T^{2} \)
5 \( 1 + (0.786 + 0.618i)T^{2} \)
11 \( 1 + (0.676 - 1.68i)T + (-0.723 - 0.690i)T^{2} \)
13 \( 1 + (0.415 - 0.909i)T^{2} \)
17 \( 1 + (0.888 + 0.458i)T^{2} \)
19 \( 1 + (0.888 - 0.458i)T^{2} \)
29 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (-0.327 + 0.945i)T^{2} \)
37 \( 1 + (0.204 - 1.06i)T + (-0.928 - 0.371i)T^{2} \)
41 \( 1 + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (-1.37 + 0.627i)T + (0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.907 + 1.75i)T + (-0.580 + 0.814i)T^{2} \)
59 \( 1 + (-0.995 + 0.0950i)T^{2} \)
61 \( 1 + (-0.981 + 0.189i)T^{2} \)
67 \( 1 + (0.348 - 0.442i)T + (-0.235 - 0.971i)T^{2} \)
71 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (-0.907 + 1.75i)T + (-0.580 - 0.814i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (0.327 + 0.945i)T^{2} \)
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

−10.14455241450566286262591128001, −9.493296379966096923543554725052, −7.983959727220755239546896699333, −7.05385170746485176260343402767, −6.15734423194763464623468822476, −5.35733236621024617884726141971, −4.65098022995546529693640781249, −3.72547905680019927366164380347, −2.69077444510185685698762129289, −1.87771475905449857482971404226, 2.50671962210506849533663171405, 3.19131600964508672476777686281, 4.14795406596289249778051397792, 5.25348859147171934997098603975, 5.83502320732120489803242746067, 6.37643506210714895045207629543, 7.62802941982746095727924241687, 8.132143897241915314142437824938, 9.018918714374520642107815177854, 10.82306679858715982909544121059

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