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Properties

Label 2-444-111.71-c0-0-0
Degree $2$
Conductor $444$
Sign $0.806 + 0.590i$
Analytic cond. $0.221584$
Root an. cond. $0.470728$
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  + (0.766 − 0.642i)3-s + (−0.326 − 0.118i)7-s + (0.173 − 0.984i)9-s + (0.0603 + 0.342i)13-s + (−0.766 + 0.642i)19-s + (−0.326 + 0.118i)21-s + (0.766 + 0.642i)25-s + (−0.500 − 0.866i)27-s − 1.87·31-s + (−0.5 + 0.866i)37-s + (0.266 + 0.223i)39-s + 1.53·43-s + (−0.673 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (0.347 + 1.96i)61-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.326 − 0.118i)7-s + (0.173 − 0.984i)9-s + (0.0603 + 0.342i)13-s + (−0.766 + 0.642i)19-s + (−0.326 + 0.118i)21-s + (0.766 + 0.642i)25-s + (−0.500 − 0.866i)27-s − 1.87·31-s + (−0.5 + 0.866i)37-s + (0.266 + 0.223i)39-s + 1.53·43-s + (−0.673 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (0.347 + 1.96i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.806 + 0.590i$
Analytic conductor: \(0.221584\)
Root analytic conductor: \(0.470728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :0),\ 0.806 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027757192\)
\(L(\frac12)\) \(\approx\) \(1.027757192\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 - 1.53T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.28067123894538320584871916573, −10.26105814562755750978931328819, −9.233245708364864765113151359932, −8.591168518192133178547845985694, −7.51545908229502291092938291488, −6.78597242917371605289112804430, −5.73189329020752339186656700527, −4.16676088245441713148138730371, −3.10975008986571417225822215838, −1.73238038472723871136137090880, 2.28585030171626943257041257353, 3.43854977385681353137818426720, 4.50506601562562782448527920410, 5.61285821749543884238413263075, 6.90529563101971354613358307839, 7.919700578568345159378711529290, 8.884734414750202435139432754886, 9.454618782170826614186488824805, 10.57462545199863649673940206141, 11.05805031418535920978730166455

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