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Properties

Label 2-1127-161.149-c0-0-0
Degree $2$
Conductor $1127$
Sign $0.358 + 0.933i$
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  + (−0.0395 − 0.829i)2-s + (0.308 − 0.0294i)4-s + (−0.154 − 1.07i)8-s + (−0.723 + 0.690i)9-s + (1.97 + 0.0941i)11-s + (−0.583 + 0.112i)16-s + (0.601 + 0.573i)18-s − 1.64i·22-s + (0.235 − 0.971i)23-s + (−0.888 − 0.458i)25-s + (0.698 + 1.53i)29-s + (−0.140 − 0.577i)32-s + (−0.202 + 0.234i)36-s + (−1.04 − 1.09i)37-s + (−0.557 − 0.0801i)43-s + (0.612 − 0.0291i)44-s + ⋯
L(s)  = 1  + (−0.0395 − 0.829i)2-s + (0.308 − 0.0294i)4-s + (−0.154 − 1.07i)8-s + (−0.723 + 0.690i)9-s + (1.97 + 0.0941i)11-s + (−0.583 + 0.112i)16-s + (0.601 + 0.573i)18-s − 1.64i·22-s + (0.235 − 0.971i)23-s + (−0.888 − 0.458i)25-s + (0.698 + 1.53i)29-s + (−0.140 − 0.577i)32-s + (−0.202 + 0.234i)36-s + (−1.04 − 1.09i)37-s + (−0.557 − 0.0801i)43-s + (0.612 − 0.0291i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1127\)    =    \(7^{2} \cdot 23\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(0.562446\)
Root analytic conductor: \(0.749964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1127} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1127,\ (\ :0),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214541928\)
\(L(\frac12)\) \(\approx\) \(1.214541928\)
\(L(1)\) 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 \( 1 + (-0.235 + 0.971i)T \)
good2 \( 1 + (0.0395 + 0.829i)T + (-0.995 + 0.0950i)T^{2} \)
3 \( 1 + (0.723 - 0.690i)T^{2} \)
5 \( 1 + (0.888 + 0.458i)T^{2} \)
11 \( 1 + (-1.97 - 0.0941i)T + (0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.327 + 0.945i)T^{2} \)
19 \( 1 + (0.327 - 0.945i)T^{2} \)
29 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.235 - 0.971i)T^{2} \)
37 \( 1 + (1.04 + 1.09i)T + (-0.0475 + 0.998i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.557 + 0.0801i)T + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.02 - 0.353i)T + (0.786 + 0.618i)T^{2} \)
59 \( 1 + (0.928 + 0.371i)T^{2} \)
61 \( 1 + (-0.723 - 0.690i)T^{2} \)
67 \( 1 + (0.833 - 1.61i)T + (-0.580 - 0.814i)T^{2} \)
71 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.981 - 0.189i)T^{2} \)
79 \( 1 + (1.02 - 0.353i)T + (0.786 - 0.618i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (-0.235 - 0.971i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−10.09074584978427728103396084721, −9.064948556845669684772150601875, −8.549460827026205324326202642841, −7.20871201546387756763108810259, −6.60254286455482351102218360356, −5.67754740005409310514935988698, −4.37653270396732222842896987927, −3.51012100811016567540112740842, −2.43790303669818373192264807247, −1.37379088010298882728531123747, 1.62476182503395644162606490316, 3.11238825089713288893966970023, 4.05214454253273264681121503821, 5.38095517895948438390949049508, 6.24912557994683553211610714236, 6.64157926198839307504170190364, 7.61210802088339021504804846460, 8.521031186388098446622653850897, 9.146844190177570536865129623246, 9.952690958311168631172850578228

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