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Properties

Label 2-279-31.15-c0-0-0
Degree $2$
Conductor $279$
Sign $0.569 - 0.822i$
Analytic cond. $0.139239$
Root an. cond. $0.373147$
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  + (−0.309 + 0.951i)4-s + (−0.5 + 1.53i)7-s + (1.11 − 1.53i)13-s + (−0.809 − 0.587i)16-s + (0.5 − 0.363i)19-s − 25-s + (−1.30 − 0.951i)28-s + (0.809 + 0.587i)31-s − 1.17i·37-s + (−0.690 − 0.951i)43-s + (−1.30 − 0.951i)49-s + (1.11 + 1.53i)52-s + 1.90i·61-s + (0.809 − 0.587i)64-s − 0.618·67-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)4-s + (−0.5 + 1.53i)7-s + (1.11 − 1.53i)13-s + (−0.809 − 0.587i)16-s + (0.5 − 0.363i)19-s − 25-s + (−1.30 − 0.951i)28-s + (0.809 + 0.587i)31-s − 1.17i·37-s + (−0.690 − 0.951i)43-s + (−1.30 − 0.951i)49-s + (1.11 + 1.53i)52-s + 1.90i·61-s + (0.809 − 0.587i)64-s − 0.618·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(0.139239\)
Root analytic conductor: \(0.373147\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 279,\ (\ :0),\ 0.569 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7017072873\)
\(L(\frac12)\) \(\approx\) \(0.7017072873\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.309 - 0.951i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + 1.17iT - T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−12.25429551024221793749060279658, −11.59438741006312579949655118428, −10.31556639387565751299516426904, −9.110512362114540777686184491711, −8.530180423166345765791809462018, −7.59159729012388780857712537145, −6.15288960651672932505622840892, −5.26886852333603508612857626278, −3.61536942286627968688796702324, −2.67822220120252157615663961840, 1.42164924081127198050690250059, 3.74259660595510231082540815675, 4.60881171644339911533000157932, 6.15755820848051759343356897152, 6.79666055717004935170590072991, 8.105114584487131635186219813774, 9.416979710571142087678992816212, 9.982823972824118449322915220139, 10.93447766632686225886904940401, 11.69898506156551839264290736672

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