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Properties

Label 2-1127-161.75-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.623 - 0.781i$
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  + (−0.759 − 1.06i)2-s + (−0.233 + 0.676i)4-s + (−0.357 + 0.105i)8-s + (−0.888 + 0.458i)9-s + (−1.11 + 1.56i)11-s + (0.946 + 0.744i)16-s + (1.16 + 0.600i)18-s + 2.51·22-s + (0.0475 + 0.998i)23-s + (−0.995 + 0.0950i)25-s + (−0.544 + 0.627i)29-s + (0.0572 − 1.20i)32-s + (−0.101 − 0.708i)36-s + (0.252 − 0.130i)37-s + (−1.61 − 0.474i)43-s + (−0.796 − 1.11i)44-s + ⋯
L(s)  = 1  + (−0.759 − 1.06i)2-s + (−0.233 + 0.676i)4-s + (−0.357 + 0.105i)8-s + (−0.888 + 0.458i)9-s + (−1.11 + 1.56i)11-s + (0.946 + 0.744i)16-s + (1.16 + 0.600i)18-s + 2.51·22-s + (0.0475 + 0.998i)23-s + (−0.995 + 0.0950i)25-s + (−0.544 + 0.627i)29-s + (0.0572 − 1.20i)32-s + (−0.101 − 0.708i)36-s + (0.252 − 0.130i)37-s + (−1.61 − 0.474i)43-s + (−0.796 − 1.11i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3312709055\)
\(L(\frac12)\) \(\approx\) \(0.3312709055\)
\(L(1)\) 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 \( 1 + (-0.0475 - 0.998i)T \)
good2 \( 1 + (0.759 + 1.06i)T + (-0.327 + 0.945i)T^{2} \)
3 \( 1 + (0.888 - 0.458i)T^{2} \)
5 \( 1 + (0.995 - 0.0950i)T^{2} \)
11 \( 1 + (1.11 - 1.56i)T + (-0.327 - 0.945i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (-0.928 - 0.371i)T^{2} \)
19 \( 1 + (-0.928 + 0.371i)T^{2} \)
29 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (-0.0475 - 0.998i)T^{2} \)
37 \( 1 + (-0.252 + 0.130i)T + (0.580 - 0.814i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.771 + 0.308i)T + (0.723 - 0.690i)T^{2} \)
59 \( 1 + (-0.235 + 0.971i)T^{2} \)
61 \( 1 + (0.888 + 0.458i)T^{2} \)
67 \( 1 + (-1.30 + 0.124i)T + (0.981 - 0.189i)T^{2} \)
71 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.786 + 0.618i)T^{2} \)
79 \( 1 + (-0.771 - 0.308i)T + (0.723 + 0.690i)T^{2} \)
83 \( 1 + (-0.415 - 0.909i)T^{2} \)
89 \( 1 + (-0.0475 + 0.998i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−10.02430273767244237909231929811, −9.667065913077801449115476254729, −8.647613982848196859458725562707, −7.920934038441842058994464882988, −7.10436265106088999681407639795, −5.71361857363057821906923170537, −5.04567402195447616592126927600, −3.64067093520010288785941612200, −2.53309733048585625828832521966, −1.80993658647095348843060977759, 0.35569124564364972450575039587, 2.68100989417336426612508572825, 3.57584301354654239556564842862, 5.20205604159405384247628096910, 5.96393206982094741803113513354, 6.47502924856946997172672576260, 7.69076479418518087556799940968, 8.246446291637665435416403681584, 8.769014914402223956110798694533, 9.640763357956622775472788993285

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