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Properties

Label 2-2156-308.163-c0-0-0
Degree $2$
Conductor $2156$
Sign $0.0809 - 0.996i$
Analytic cond. $1.07598$
Root an. cond. $1.03729$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)18-s + (0.809 + 0.587i)22-s + (1.01 + 0.587i)23-s + (−0.913 − 0.406i)25-s + (0.5 + 0.363i)29-s + (0.499 + 0.866i)32-s + (−0.809 − 0.587i)36-s + (1.47 − 0.658i)37-s + 1.90i·43-s + (−0.5 + 0.866i)44-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)18-s + (0.809 + 0.587i)22-s + (1.01 + 0.587i)23-s + (−0.913 − 0.406i)25-s + (0.5 + 0.363i)29-s + (0.499 + 0.866i)32-s + (−0.809 − 0.587i)36-s + (1.47 − 0.658i)37-s + 1.90i·43-s + (−0.5 + 0.866i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.0809 - 0.996i$
Analytic conductor: \(1.07598\)
Root analytic conductor: \(1.03729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :0),\ 0.0809 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214350200\)
\(L(\frac12)\) \(\approx\) \(1.214350200\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 \)
11 \( 1 + (-0.669 + 0.743i)T \)
good3 \( 1 + (-0.669 - 0.743i)T^{2} \)
5 \( 1 + (0.913 + 0.406i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.104 - 0.994i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T^{2} \)
23 \( 1 + (-1.01 - 0.587i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.913 + 0.406i)T^{2} \)
37 \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.913 + 0.406i)T^{2} \)
67 \( 1 + (-1.64 + 0.951i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (1.41 - 1.27i)T + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−9.390653055162438620885447152231, −8.468255155911914041637762691564, −7.85032774459905379509427940607, −7.13234202738077366663076337299, −6.33076629989246699792657203148, −5.61966512870565069430697288674, −4.68861280145512400422675370761, −4.01531205183555088573929775424, −2.92486189706581656547484522016, −1.26683188501870678667768599857, 1.06641710939528831820378112154, 2.13443325513619328430441501857, 3.26943665264599669528267914914, 4.14003316636666484513030518336, 4.71928452536519784101442510073, 5.84010297581943704800357386795, 6.72695083850832415561837108611, 7.56290744372002735875204274828, 8.627010823943086648310666178849, 9.270622441915439542975924100235

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